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

DOI:
https://doi.org/10.1090/S0025-5718-1985-0804944-8

MathSciNet review:
804944

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Abstract | References | Similar Articles | Additional Information

Abstract: Consider the sequence generated by , , where , , 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 defined by , the resulting sequence is a subsequence of . Second, if Newton's method and the secant method are used (with suitable starting values) to solve the equation , then the sequences obtained from both of those methods are also subsequences of the original sequence.

**[1]**G. A. Baker & P. R. Graves-Morris,*Padé Approximants, Part*1,*Basic Theory*, Vol. 13, Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, Mass., 1981. MR**635619 (83a:41009a)****[2]**J. Gill & G. Miller, "Newton's method and ratios of Fibonacci numbers,"*Fibonacci Quart.*, v. 19, 1981, pp. 1-4. MR**606100 (82h:65090)****[3]**V. E. Hoggatt,*Fibonacci and Lucas Numbers*, Houghton Mifflin, Boston, Mass., 1969.**[4]**E. Isaacson & H. B. Keller,*Analysis of Numerical Methods*, Wiley, New York, 1966. MR**0201039 (34:924)****[5]**G. M. Phillips, "Aitken sequences and Fibonacci numbers,"*Amer. Math. Monthly*, v. 91, 1984, pp. 354-357. MR**750521 (85h:65013)****[6]**N. N. Vorob'ev,*Fibonacci Numbers*, Popular Lectures in Mathematics Series, Vol. 2 (transl. from Russian), Pergamon Press, Oxford, 1961. MR**0146138 (26:3664)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0804944-8

Keywords:
Fibonacci sequence,
Aitken acceleration,
Newton's method,
secant method,
Padé approximation,
continued fraction

Article copyright:
© Copyright 1985
American Mathematical Society