On Newton's Method and Rational Approximations to Quadratic Irrationals

http://dx.doi.org/10.4153/CMB-2004-002-4
Canad. Math. Bull. 47(2004), 12-16

  Published:2004-03-01
 Printed: Mar 2004

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.