Aitken sequences and generalized Fibonacci numbers
J. H. McCabe and G. M. Phillips
Math. Comp. 45 (1985), 553-558
Primary 41A21; Secondary 11B39, 65B05
Full-text PDF Free Access
Similar Articles |
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.
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
Gill and Gary
Miller, Newton’s method and ratios of Fibonacci numbers,
Fibonacci Quart. 19 (1981), no. 1, 1–4. MR 606100
V. E. Hoggatt, Fibonacci and Lucas Numbers, Houghton Mifflin, Boston, Mass., 1969.
Isaacson and Herbert
Bishop Keller, Analysis of numerical methods, John Wiley &
Sons Inc., New York, 1966. MR 0201039
M. Phillips, Aitken sequences and Fibonacci numbers, Amer.
Math. Monthly 91 (1984), no. 6, 354–357. MR 750521
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
- 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)
- J. Gill & G. Miller, "Newton's method and ratios of Fibonacci numbers," Fibonacci Quart., v. 19, 1981, pp. 1-4. MR 606100 (82h:65090)
- V. E. Hoggatt, Fibonacci and Lucas Numbers, Houghton Mifflin, Boston, Mass., 1969.
- E. Isaacson & H. B. Keller, Analysis of Numerical Methods, Wiley, New York, 1966. MR 0201039 (34:924)
- G. M. Phillips, "Aitken sequences and Fibonacci numbers," Amer. Math. Monthly, v. 91, 1984, pp. 354-357. MR 750521 (85h:65013)
- N. N. Vorob'ev, Fibonacci Numbers, Popular Lectures in Mathematics Series, Vol. 2 (transl. from Russian), Pergamon Press, Oxford, 1961. MR 0146138 (26:3664)
Retrieve articles in Mathematics of Computation
Retrieve articles in all journals
© Copyright 1985 American Mathematical Society