> > Function F(x) does not have a formula to specify the relation between x > > and f(x), but for each x, I have got the value of f(x) by a measurement > > machine. > > > > I am interested in the point xi at which the second derivative of f(xi) > > is zero. > > > > Can anybody help me how to do that? > > Luc made good recommendations for simple methods using low order polynomials and finite differences.
I would suggest that you are unlikely to be happy with the results of either approach due to the noise introduced by the measurement machine. To see the problem try substituting f(x) + noise into the finite difference equations and watch how the noise can dominate the computation. Other forms of error such as round-off can be even worse. The only reasonably good answer that I know if a variant on Luc's polynomials and involves computing an empirical distribution of possible models for you data and then from that distribution getting a distribution of possible answers. Gaussian processes are a very nice way to do this but the methods are not particularly simple.
