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- W4241009969 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 Halleyâs method, to elements of ${ {r_n}}$ and ${ {R_n}}$ is also investigated." @default.
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- W4241009969 date "1993-07-01" @default.
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- W4241009969 title "Generalized Fibonacci and Lucas Sequences and Rootfinding Methods" @default.
- W4241009969 doi "https://doi.org/10.2307/2152961" @default.
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