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On Newton’s method and rational approximations to quadratic irrationals. (English) Zbl 1080.11007

In 1999, G. J. Rieger [Fibonacci Q. 37, No. 2, 178–179 (1999; Zbl 0943.11004)] exhibited a differentiable function \(f\) having \(q=(-1+\sqrt 5)/2\) as a root and enjoying the following property: the approximations generated by Newton’s method, \(x_{n+1}=x_n-f(x_n)/f' (x_n)\) (with \(n\geq 0\) and \(x_0=0)\), are given by \(x_n=\frac{p_{2n}} {q_{2n}}=\frac{F_{2n}} {F_{2n+1}}\), viz. the even-indexed convergents of the golden ratio. In 2001, T. Komatsu [Fibonacci Q. 39, No. 4, 336–338 (2001; Zbl 0992.11008)] extended the result to quadratic irrationals given by \([0,\overline{a,b}]\). In the note under review, E. B. Burger generalizes these results to quadratic irrationalities of the form \(\alpha=[0,\overline{a_1,a_2,\dots,a_L}]\). The result is beautiful and the paper is clearly written.

MSC:

11A55 Continued fractions
11J70 Continued fractions and generalizations
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