Householder's approximants and continued fraction expansion of quadratic irrationals (CROSBI ID 587865)
Prilog sa skupa u zborniku | sažetak izlaganja sa skupa | međunarodna recenzija
Podaci o odgovornosti
Petričević, Vinko
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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Podaci o prilogu
75-75.
2012.
objavljeno
Podaci o matičnoj publikaciji
15th International Conference on Fibonacci Numbers and Their Applications
Liptai, Kalman
Eger: Eszterhazy Karoly College
Podaci o skupu
15th International Conference on Fibonacci Numbers and Their Applications
predavanje
25.06.2012-30.06.2012
Eger, Mađarska