On Newton's Method and Rational Approximations to Quadratic Irrationals

[摘要] In 1988 Rieger exhibited a differentiable function having a zero atthe golden ratioreak$(-1+sqrt5)/2$ for which when Newton's method for approximatingroots is applied with an initial value $x_0=0$, all approximatesare so-called ``best rational approximates''---in this case, of theform $F_{2n}/F_{2n+1}$, where $F_n$ denotes the $n$-th Fibonaccinumber. Recently this observation was extended by Komatsu to theclass of all quadratic irrationals whose continued fractionexpansions have period length $2$. Here we generalize theseobservations by producing an analogous result for all quadraticirrationals and thus provide an explanation for these phenomena.

[学科分类] 数学（综合）

[关键词] Neumann Laplacian;horn-like domain;spectrum