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This work investigates finite differences and the use of interpolation modelsto obtain approximations to the first and second derivatives of a function.Here, it is shown that if a particular set of points is used in theinterpolation model, then the solution to the associated linear system (i.e.,approximations to the gradient and diagonal of the Hessian) can be obtained in$\mathcal{O}(n)$ computations, which is the same cost as finite differences,and is a saving over the $\mathcal{O}(n^3)$ cost when solving a generalunstructured linear system. Moreover, if the interpolation points are formedusing a `regular minimal positive basis', then the error bound for the gradientapproximation is the same as for a finite differences approximation. Numericalexperiments are presented that show how the derivative estimates can beemployed within an existing derivative free optimization algorithm, thusdemonstrating one of the potential practical uses of these derivativeapproximations.

Gradient and Hessian approximations in Derivative Free Optimization