Generalized Fibonacci and Lucas sequences and rootfinding methods

Author:
Joseph B. Muskat

Journal:
Math. Comp. **61** (1993), 365-372

MSC:
Primary 65B99; Secondary 11B39, 49M15, 65H99, 90C30

DOI:
https://doi.org/10.1090/S0025-5718-1993-1192974-3

MathSciNet review:
1192974

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Abstract: Consider the sequences and generated by and , where , with *p* and *q* real and nonzero. The Fibonacci sequence and the Lucas sequence are special cases of and , respectively. Define , where *d* is a positive integer. McCabe and Phillips showed that for , applying one step of Aitken acceleration to any appropriate triple of elements of yields another element of . They also proved for that if a step of the Newton-Raphson method or the secant method is applied to elements of in solving the characteristic equation , then the result is an element of .

The above results are obtained for . It is shown that if any of the above methods is applied to elements of , then the result is an element of . The application of certain higher-order iterative procedures, such as Halley's method, to elements of and is also investigated.

**[1]**W. Gander,*On Halley's iteration method*, Amer. Math. Monthly**92**(1985), 131-134. MR**777559 (86f:65094)****[2]**L. E. Dickson,*History of the theory of numbers, Vol.*1, Chelsea, New York, 1952.**[3]**A. F. Horadam,*Basic properties of a certain generalized sequence of numbers*, Fibonacci Quart.**3**(1965), 161-176. MR**0186615 (32:4074)****[4]**M. J. Jamieson,*Fibonacci numbers and Aitken sequences revisited*, Amer. Math. Monthly**97**(1990), 829-831. MR**1080391 (92f:11026)****[5]**D. H. Lehmer,*A machine method for solving polynomial equations*, J. Assoc. Comput. Mach.**8**(1961), 151-162.**[6]**J. H. McCabe and G. M. Phillips,*Aitken sequences and generalized Fibonacci numbers*, Math. Comp.**45**(1985), 553-558. MR**804944 (87b:41015)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1993-1192974-3

Article copyright:
© Copyright 1993
American Mathematical Society