Continued fractions and Newton's approximations. II (Q2746562)

Let \(p_n/q_n\) be the convergents of the simple continued fraction expansion of \(\theta= \frac{1}{2a} (\sqrt{D}-ab)= [0, \overline{a,b}]\), where \(D= ab(ab+4)\) with positive integers \(a\) and \(b\). The author constructs a decreasing convex function \(H\) with values \(H(0)= 1\), \(H(\theta)= 0\), such that the sequence \(x_0= 0\), \(x_{n+1}= x_n- H(x_n)/ H'(x_n)\) of Newton's approximations for \(\theta\) yields the convergents \(x_n= p_{2n}/ q_{2n}\). The case \(a=b=1\) of the golden section \(\theta= \frac{1}{2} (\sqrt{5}-1)\) was treated by \textit{G. J. Rieger} [Fibonacci Q. 37, 178-179 (1999; Zbl 0943.11004)]. In a previous paper [Math. Commun. 4, 167-176 (1999; Zbl 0972.11008)] the author dealt with \(\sqrt{a^2\pm 4}\).