## On subsequences of convergents to a quadratic irrational given by some numerical schemes.(English)Zbl 1223.11087

Given a quadratic irrational $$\alpha$$, the author is interested in how some numerical schemes (secant-like methods, the false position method, Newton’s method ) applied to a convenient function $$f$$ provide subsequences of convergents to $$\alpha$$. For a large class of $$\alpha$$, starting with suitable $$x_0$$ and $$x_1$$, he gets a subsequence of convergents to $$\alpha$$ of the form $$\frac{p_{F_n}}{q_{F_n}}$$ where $$F_n$$ verify $$F_n=F_{n-1}+F_{n-2}+z_n$$ where $$z_n$$ is a bounded sequence and other subsequences of convergents given by linear recurring sequences. The method of false positions gives arithmetical subsequences of convergents. He shows an explicit way to construct a function $$f_\alpha$$ and an initial value $$x_0$$ for which Newton’s formula gives exactly the convergents of the sequence $$\frac{p_{nL+k}}{q_{nL+k}}$$, where $$k$$ is any integer and $$L$$ the length of any period of the partial quotients in the continued fraction expansion of $$\alpha$$.

### MSC:

 11J70 Continued fractions and generalizations
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### References:

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