Naive Taylor approximations

For the past week Jan has been considering various ways to implemented Taylor approximations for real vector functions. While automatic differentiation has seemed a natural choice, the issue of computing safe approximations of the interval remainder makes that choice questionable, as it requires the re-computation of all Taylor coefficients of lower orders. With the requirement of a rigorous Taylor approximation, symbolic differentiation becomes a viable alternative. The large degree of sharing between terms of a Taylor approximation makes is crucial, from a performance perspective, to memoize the intermediate value in the computation. However, the optimization space is very large and Jan decided to aim to complete a prototype as soon as possible, to be able to observe the expected  behavior of the Lohner algorithm, and thus motivate the considerable optimization work that will be needed to implement the algorithm in a satisfactory manner. Jan has finalized a naive implementation of taylor approximation for multivariate functions and will aim for an implementation of Taylor methods for ODEs next.