## Continued fractions and Newton’s approximations. II.(English)Zbl 0992.11008

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 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 $$\sqrt{a^2\pm 4}$$.

### MSC:

 11A55 Continued fractions

### Citations:

Zbl 0943.11004; Zbl 0972.11008