Application of a Generalized Secant Method to Nonlinear Equations with Complex Roots

The secant method is a very effective numerical procedure used for solving nonlinear equations of the form $f(x)=0$. In a recent work [A. Sidi, Generalization of the secant method for nonlinear equations. {\em Appl. Math. E-Notes}, 8:115--123, 2008] we presented a generalization of the secant method that uses only one evaluation of $f(x)$ per iteration, and we provided a local convergence theory for it that concerns real roots. For each integer $k$, this method generates a sequence $\{x_n\}$ of approximations to a real root of $f(x)$, where, for $n\geq k$, $x_{n+1}=x_n-f(x_n)/p'_{n,k}(x_n)$, $p_{n,k}(x)$ being the polynomial of degree $k$ that interpolates $f(x)$ at $x_n,x_{n-1},\ldots,x_{n-k}$, the order $s_k$ of this method satisfying $1<s_k<2$. Clearly, when $k=1$, this method reduces to the secant method with $s_1=(1+\sqrt{5})/2$. In addition, $s_1<s_2<s_3<\cdots,$ such that and $\lim_{k\to\infty}s_k=2$. In this note, we study the application of this method to simple complex roots of a real or complex function $f(z)$. We show that the local convergence theory developed for real roots can be extended almost as is to complex roots, provided suitable assumptions and justifications are made. We illustrate the theory with two numerical examples.