On Newton's Method and Rational Approximations to Quadratic Irrationals

In 1988 Rieger exhibited a differentiable function having a zero at the golden ratio (−1 + )/2 for which when Newton's method for approximating roots is applied with an initial value x0 = 0, all approximates are so-called “best rational approximates”—in this case, of the form F2n/F2n+1, where Fn 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.