  This paper presents two methods for approximating a proper subset of the
entries of a Hessian using only function evaluations. These approximations are
obtained using the techniques called \emph{generalized simplex Hessian} and
\emph{generalized centered simplex Hessian}. We show how to choose the matrices
of directions involved in the computation of these two techniques depending on
the entries of the Hessian of interest. We discuss the number of function
evaluations required in each case and develop a general formula to approximate
all order-$P$ partial derivatives. Since only function evaluations are required
to compute the methods discussed in this paper, they are suitable for use in
derivative-free optimization methods.
