Geodesic
On Newton's Method and Rational Approximations to Quadratic Irrationals
Canadian mathematical bulletin, Tome 47 (2004) no. 1, pp. 12-16

Voir la notice de l'article provenant de la source Cambridge University Press

In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio $(-1\,+\,\sqrt{5})/2$ for which when Newton's method for approximating roots is applied with an initial value ${{x}_{0}}\,=\,0$ , all approximates are so-called “best rational approximates”—in this case, of the form ${{F}_{2n}}/{{F}_{2n+1}}$ , where ${{F}_{n}}$ denotes the $n$ -th Fibonacci number. Recently this observation was extended by Komatsu to the class of all quadratic irrationals whose continued fraction expansions have period length 2. Here we generalize these observations by producing an analogous result for all quadratic irrationals and thus provide an explanation for these phenomena.
DOI : 10.4153/CMB-2004-002-4
Mots-clés : 11A55, 11B37 
Burger, Edward B. On Newton's Method and Rational Approximations to Quadratic Irrationals. Canadian mathematical bulletin, Tome 47 (2004) no. 1, pp. 12-16. doi: 10.4153/CMB-2004-002-4
@article{10_4153_CMB_2004_002_4,
     author = {Burger, Edward B.},
     title = {On {Newton's} {Method} and {Rational} {Approximations} to {Quadratic} {Irrationals}},
     journal = {Canadian mathematical bulletin},
     pages = {12--16},
     year = {2004},
     volume = {47},
     number = {1},
     doi = {10.4153/CMB-2004-002-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2004-002-4/}
}
TY  - JOUR
AU  - Burger, Edward B.
TI  - On Newton's Method and Rational Approximations to Quadratic Irrationals
JO  - Canadian mathematical bulletin
PY  - 2004
SP  - 12
EP  - 16
VL  - 47
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2004-002-4/
DO  - 10.4153/CMB-2004-002-4
ID  - 10_4153_CMB_2004_002_4
ER  - 
%0 Journal Article
%A Burger, Edward B.
%T On Newton's Method and Rational Approximations to Quadratic Irrationals
%J Canadian mathematical bulletin
%D 2004
%P 12-16
%V 47
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2004-002-4/
%R 10.4153/CMB-2004-002-4
%F 10_4153_CMB_2004_002_4

[1] [1] Burger, E. B., Exploring the Number Jungle: A Journey into Diophantine Analysis. Amer.Math. Soc., Student Mathematical Library Series, Providence, (2000). Google Scholar

[2] [2] Komatsu, T., Continued fractions and Newton's approximations. Math. Commun. 4 (1999), 167–176. Google Scholar

[3] [3] Komatsu, T., Continued fractions and Newton's approximations, II. Fibonacci Quart. 39 (2001), 336–338. Google Scholar

[4] [4] Rieger, G. J., The golden section and Newton's approximation. Amer.Math. Soc. Abstracts 9(1988), 88T-11-244. Google Scholar

[5] [5] Rieger, G. J., The golden section and Newton's approximation. Fibonacci Quart. 37 (1999), 178–179. Google Scholar

[6] [6] van der Poorten, A. J., An introduction to continued fractions. In: Diophantine Analysis, (eds., J. H. Loxton and A. J. van der Poorten), LondonMath. Soc. Lecture Note Series 109, Cambridge University Press, (1986), 99–138. Google Scholar

Cité par Sources :