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This work presents a novel matrix-based method for constructing an approximation Hessian using only function evaluations. The method requires less computational power than interpolation-based methods and is easy to implement in matrix-based programming languages such as MATLAB. As only function evaluations are required, the method is suitable for use in derivative-free algorithms. For reasonably structured sample sets, the method is proven to create an order- |$1$| accurate approximation of the full Hessian. Under more specialized structures, the method is proved to yield order- |$2$| accuracy. The underdetermined case, where the number of sample points is fewer than required for full interpolation, is studied and error bounds are developed for the resulting partial Hessians. [ABSTRACT FROM AUTHOR]

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]