  Using the Moore--Penrose pseudoinverse, this work generalizes the gradient
approximation technique called centred simplex gradient to allow sample sets
containing any number of points. This approximation technique is called the
\emph{generalized centred simplex gradient}. We develop error bounds and, under
a full-rank condition, show that the error bounds have order $O(\Delta^2)$,
where $\Delta$ is the radius of the sample set of points used. We establish
calculus rules for generalized centred simplex gradients, introduce a
calculus-based generalized centred simplex gradient and confirm that error
bounds for this new approach are also order $O(\Delta^2)$. We provide several
examples to illustrate the results and some benefits of these new methods.
