Technical Report Nr. It is based on approriate representations of the gradient and the Hessian. After a brief exposition of the forward and reverse mode of automatic differentiation for derivatives up to second order and compact proofs of their complexities, the new forward mode is presented and analyzed. Except for the fact that the new method is a very efficient implementation of the forward mode, it can be used to significantly reduce the length of characterizing sequences before applying the memory expensive reverse mode. For the Hessian all discussed algorithms are shown to be efficiently parallelizable.
Some numerical examples confirm the advantages of the new forward mode.