In this paper we study how to accelerate the convergence of the ratios (x_n) of generalized Fibonacci sequences. In particular, we provide recurrent formulas in order to generate subsequences (x_g_n) for every linear recurrent sequence (g_n) of order 2. Using these formulas we prove that some approximation methods, as secant, Newton, Halley and Householder methods, can generate subsequences of (xn). Moreover, interesting properties on Fibonacci numbers arise as an application. Finally, we apply all the results to the convergents of a particular continued fraction which represents quadratic irrationalities.

Accelerations of Generalized Fibonacci Sequences

CERRUTI, Umberto;MURRU, NADIR
2011

Abstract

In this paper we study how to accelerate the convergence of the ratios (x_n) of generalized Fibonacci sequences. In particular, we provide recurrent formulas in order to generate subsequences (x_g_n) for every linear recurrent sequence (g_n) of order 2. Using these formulas we prove that some approximation methods, as secant, Newton, Halley and Householder methods, can generate subsequences of (xn). Moreover, interesting properties on Fibonacci numbers arise as an application. Finally, we apply all the results to the convergents of a particular continued fraction which represents quadratic irrationalities.
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http://arxiv.org/pdf/1301.3477.pdf
http://www.fq.math.ca/
Fibonacci sequences; approximation methods; secant; Newton; Halley; Householder
Marco Abrate; Stefano Barbero; Umberto Cerruti; Nadir Murru
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2318/95536
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