Computing derivatives in optimisation

Newton’s method for function minimisation possesses an outstanding property. Its final quadratic convergence under very weak hypothesis, but also some drawbacks, among which (i) the possibility of non convergence when starting far from the solution sought and (ii) the necessity of computing (and then programming) the first and second partial derivatives. This paper shows how to avoid both difficulties, assuming however that the first partial derivatives are computed by exact formulae. The second partial derivatives are computed by central differences of first derivatives. The difference step h is calculated so as to establish a trade-off between accuracy and computation speed. Encouraging numerical experiments support the proposed method.