Efficient Taylor expansion computation of multidimensional vector functions on GPU

Abstract

The Taylor expansion [19] is used in many applications for a value estimation of scalar functions of one or two variables in the neighbour point. Usually, only the first two elements of the Taylor expansion are used, i.e. a value in the given point and derivatives estimation. The Taylor expansion can be also used for vector functions, too. The usual formulae are well known, but if the second element of the expansion, i.e. with the second derivatives are to be used, mathematical formulations are getting too complex for efficient programming, as it leads to the use of multi-dimensional matrices. This contribution describes a new form of the Taylor expansion for multidimensional vector functions. The proposed approach uses "standard" formalism of linear algebra, i.e. using vectors and matrices, which is simple, easy to implement. It leads to efficient computation on the GPU in the three dimensional case, as the GPU offers fast vector-vector computation and many parts can be done in parallel.