engleski

Householder's approximants and continued fraction expansion of quadratic irrationals

Let $\alpha$ be a quadratic irrational. It is well known that the continued fraction expansion of $\alpha$ is periodic. We observe Householder's approximant of order $m-1$ for the equation $(x-\alpha)(x-\alpha')=0$ and $x_0=p_n/q_n$: $R^{; ; (m)}; ; _n = \frac{; ; \alpha(p_n/q_n-\alpha')^{; ; m}; ; - \alpha' (p_n/q_n-\alpha)^{; ; m}; ; }; ; {; ; (p_n/q_n-\alpha')^{; ; m}; ; - (p_n/q_n-\alpha)^{; ; m}; ; }; ; $. We say that $R^{; ; (m)}; ; _n$ is good approximant if $R^{; ; (m)}; ; _n$ is a convergent of $\alpha$. When period begins with $a_1$, there is a good approximant at the end of the period, and when period is palindromic and has even length $\ell$, there is a good approximant in the half of the period. So when $\ell\le2$, then every approximant is good, and then it holds $R^{; ; (m)}; ; _n=\frac{; ; p_{; ; m(n+1)-1}; ; }; ; {; ; q_{; ; m(n+1)-1}; ; }; ; $ for all $n\ge0$. We prove that to be a good approximant is the palindromic and the periodic property. Further, we define the numbers $j^{; ; (m)}; ; =j^{; ; (m)}; ; (\alpha, n)$ by $R^{; ; (m)}; ; _n=\frac{; ; p_{; ; m(n+1)-1+2j}; ; }; ; {; ; q_{; ; m(n+1)-1+2j}; ; }; ; $ if $R^{; ; (m)}; ; _n$ is a good approximant. We prove that $|j^{; ; (m)}; ; |$ is unbounded by constructing an explicit family of quadratic irrationals, which involves the Fibonacci numbers.

Continued fractions; Householder's iterative methods

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