Numeric-Symbolic Composite Derivative Calculations

Abstract

 Composite derivative calculations arise in many applications in computational science and engineering.  Since 1857 the gold standard for computing composite derivatives is the celebrated formula of Faà Di Bruno.  The equation is an identity for generalizing the chain rule of calculus to higher dimensions.  It is very complicated.  Each sub calculation must satisfy two integer constraint equations. An alternative problem formulation, proposed in 1861 by George Scott, is analytically very simple: nevertheless, the requirement for computing hand-generated complex derivatives while enforcing a boundary condition, has limited its application.  Symbolic methods are also available, but computationally expensive to embed in application software.  This paper combines the best features of symbolic processing and Scott’s formulation.  The symbolic preprocessor  computes (1) derivatives, and (2) enforces the derivative boundary condition appearing in Scott’s method.  For n requested composite derivatives; the preprocessor generates a lower triangular nxn array that is embedded in the application software for computing the numerical composite derivatives.  Unless the number of requested composite work derivatives increases, the preprocessor is only called one time.  The symbolic preprocessor easily scales for handling ten’s to hundred’s of composite derivatives.  A numerical example is provided, where 1..5 composite derivatives are computed.