Naslov
Householder's approximants and continued fraction expansion of quadratic irrationals

Autori
Petričević, Vinko

Vrsta, podvrsta i kategorija rada
Sažeci sa skupova, sažetak, znanstveni

Izvornik
15th International Conference on Fibonacci Numbers and Their Applications / Liptai, Kalman - Eger : Eszterhazy Karoly College, 2012, 75-75

Skup
15th International Conference on Fibonacci Numbers and Their Applications

Mjesto i datum
Eger, Mađarska, 25.06.2012. - 30.06.2012

Vrsta sudjelovanja
Predavanje

Vrsta recenzije
Međunarodna recenzija

Ključne riječi
Continued fractions ; Householder's iterative methods

Sažetak
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.

Izvorni jezik
Engleski

Znanstvena područja
Matematika

POVEZANOST RADA