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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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 $ \{ {u_n}\} $ and $ \{ {v_n}\} $ generated by $ {u_{n + 1}} = p{u_n} - q{u_{n - 1}}$ and $ {v_{n + 1}} = p{v_n} - q{v_{n - 1}},n \geq 1$, where $ {u_0} = 0,{u_1} = 1,{v_0} = 2,{v_1} = p$ , with p and q real and nonzero. The Fibonacci sequence and the Lucas sequence are special cases of $ \{ {u_n}\} $ and $ \{ {v_n}\} $, respectively. Define $ {r_n} = {u_{n + d}}/{u_n},{R_n} = {v_{n + d}}/{v_n}$, where d is a positive integer. McCabe and Phillips showed that for $ d = 1$, applying one step of Aitken acceleration to any appropriate triple of elements of $ \{ {r_n}\} $ yields another element of $ \{ {r_n}\} $. They also proved for $ d = 1$ that if a step of the Newton-Raphson method or the secant method is applied to elements of $ \{ {r_n}\} $ in solving the characteristic equation $ {x^2} - px + q = 0$, then the result is an element of $ \{ {r_n}\} $.

The above results are obtained for $ d > 1$. It is shown that if any of the above methods is applied to elements of $ \{ {R_n}\} $, then the result is an element of $ \{ {r_n}\} $. The application of certain higher-order iterative procedures, such as Halley's method, to elements of $ \{ {r_n}\} $ and $ \{ {R_n}\} $ is also investigated.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1993-1192974-3
PII: S 0025-5718(1993)1192974-3
Article copyright: © Copyright 1993 American Mathematical Society