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Aitken sequences and generalized Fibonacci numbers

Authors: J. H. McCabe and G. M. Phillips
Journal: Math. Comp. 45 (1985), 553-558
MSC: Primary 41A21; Secondary 11B39, 65B05
MathSciNet review: 804944
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Abstract: Consider the sequence $ ({v_n})$ generated by $ {v_{n + 1}} = a{v_n} - b{v_{n - 1}}$, $ n \geqslant 2$, where $ {v_1} = 1$, $ {v_2} = a$, with a and b real, of which the Fibonacci sequence is a special case. It is shown that if Aitken acceleration is used on the sequence $ ({x_n})$ defined by $ {x_n} = {v_{n + 1}}/{v_n}$, the resulting sequence is a subsequence of $ ({x_n})$. Second, if Newton's method and the secant method are used (with suitable starting values) to solve the equation $ {x^2} - ax + b = 0$, then the sequences obtained from both of those methods are also subsequences of the original sequence.

References [Enhancements On Off] (What's this?)

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Keywords: Fibonacci sequence, Aitken acceleration, Newton's method, secant method, Padé approximation, continued fraction
Article copyright: © Copyright 1985 American Mathematical Society

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