  The secant method is a very effective numerical procedure used for solving
nonlinear equations of the form $f(x)=0$. It is derived via a linear
interpolation procedure and employs only values of $f(x)$ at the approximations
to the root of $f(x)=0$, hence it computes $f(x)$ only once per iteration. In
this note, we generalize it by replacing the relevant linear interpolant by a
suitable $(k+1)$-point polynomial of interpolation, where $k$ is an integer at
least 2. Just as the secant method, this generalization too enjoys the property
that it computes $f(x)$ only once per iteration. We provide its error in closed
form and analyze its order of convergence $s_k$. We show that this order of
convergence is greater than that of the secant method, and it increases towards
$2$ as $k\to \infty$. (Indeed, $s_7=1.9960\cdots$, for example.) This is true
for the efficiency index of the method too. We also confirm the theory via an
illustrative example.
