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 | 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]**George A. Baker Jr. and Peter Graves-Morris,*Padé approximants. Part I*, Encyclopedia of Mathematics and its Applications, vol. 13, Addison-Wesley Publishing Co., Reading, Mass., 1981. Basic theory; With a foreword by Peter A. Carruthers. MR**635619****[2]**John Gill and Gary Miller,*Newton’s method and ratios of Fibonacci numbers*, Fibonacci Quart.**19**(1981), no. 1, 1–4. MR**606100****[3]**V. E. Hoggatt,*Fibonacci and Lucas Numbers*, Houghton Mifflin, Boston, Mass., 1969.**[4]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[5]**G. M. Phillips,*Aitken sequences and Fibonacci numbers*, Amer. Math. Monthly**91**(1984), no. 6, 354–357. MR**750521**, 10.2307/2322139**[6]**N. N. Vorob′ev,*Fibonacci numbers*, Translated from the Russian by Halina Moss; translation editor Ian N. Sneddon, Blaisdell Publishing Co. (a division of Random House), New York-London, 1961. MR**0146138**

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Additional Information

DOI:
http://dx.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