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# On Newton's Method and Rational Approximations to Quadratic Irrationals

Published:2004-03-01
Printed: Mar 2004
• Edward B. Burger
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## Abstract

In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio\break $(-1+\sqrt5)/2$ 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.
 MSC Classifications: 11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11B37 - Recurrences {For applications to special functions, see 33-XX}