  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.
