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

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.

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

Article copyright:
© Copyright 1993
American Mathematical Society