On subsequences of convergents to a quadratic irrational given by some numerical schemes (Q628843)

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 \( \alpha\).