Description:

• autodiff computes the first n derivatives of f at point x0. The computation is performed numerically, without symbolically differentiating the expression of f. Yet, the computation is safe since small interval enclosures are produced. More precisely, autodiff returns a list [f_0, ..., f_n] such that, for each i, f_i is a small interval enclosing the exact value of f^(i)(x0).
• Since it does not perform any symbolic differentiation, autodiff is much more efficient than diff and should be prefered when only numerical values are necessary.
• If an interval I is provided instead of a point x0, the list returned by autodiff satisfies: for each i, f^(i)(I) is included in f_i. A particular use is when one wants to know the successive derivatives of a function at a non representable point such as pi. In this case, it suffices to call autodiff with the (almost) point interval I = [pi].
• When I is almost a point interval, the returned enclosures f_i are also almost point intervals. However, when the interval I begins to be fairly large, the enclosures can be deeply overestimated due to the dependecy phenomenon present with interval arithmetic.
• As a particular case, f_0 is an enclosure of the image of f over I. However, since the algorithm is not specially designed for this purpose it is not very efficient for this particular task. In particular, it is not able to return a finite enclosure for functions with removable singularities (e.g. sin(x)/x at 0). The command evaluate is much more efficient for computing an accurate enclosure of the image of a function over an interval.