Generalized Fibonacci and Lucas sequences and rootfinding methods

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.