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The secant method is a very effective numerical procedure used for solvingnonlinear 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 methodthat uses only one evaluation of $f(x)$ per iteration, and we provided a localconvergence theory for it that concerns real roots. For each integer $k$, thismethod 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 methodto simple complex roots of a real or complex function $f(z)$. We show that thelocal convergence theory developed for real roots can be extended almost as isto complex roots, provided suitable assumptions and justifications are made. Weillustrate the theory with two numerical examples.

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