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

Citations:

Zbl 0943.11004; Zbl 0992.11008
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