An algorithm for numerical differentiation of a function of one real variable

An Algol 60 procedure is described which will estimate the first, second or third order derivative of a function f(x) at a point x₀, using polynomial interpolation to values of f(x) at points on the real axis. This procedure does not require the user to provide information about the accuracy of the function values. The procedure itself monitors the noise level in these values, and allows for the effect of noise on the estimated derivative. The evaluation points may be restricted to one side of x₀ if f(x) makes this necessary, and are chosen either to minimise the error in the result, or to achieve a specified tolerance with as few function evaluations as possible.