Efficient Numerical Evaluation of Arbitrary High-order Partial Derivatives via Algebraic Decomposition

Abstract

Computing high-order partial derivatives of multivariate functions is often computationally and memory intensive. This paper addresses this problem via an algebraic approach. High-order partial derivatives of a multivariate function are expressed as linear combinations of univariate directional derivatives, which are computationally friendly. The coefficients and direction vectors are determined by the Waring decomposition of monomials, so that the number of univariate directional derivatives in the representation is minimized. Numerical experiments confirm the efficiency of the proposed method.