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