Symbolic vs. Algorithmic Differentiation of GSL Integration Routines

Forward and reverse modes of algorithmic differentiation (AD) transform implementations of multivariate vector functions F : IR → IR as computer programs into tangent and adjoint code, respectively. The adjoint mode is of particular interest in large-scale functions due to the independence of its computational cost on the number of free variables. The additional memory requirement for the computation of derivatives of the output with respect to parameters by a fully algorithmic method (derived by AD) can quickly become prohibitive for large values of n. This can be reduced significantly by the symbolic approach to differentiation of the underlying integration routine. Vectorizing gsl routines for integration and applying symbolic adjoint on them has considerably less memory requirement with nearly the same runtime overhead and in most cases faster convergence in comparison with algorithmic adjoint. 1 Differentiation of Integrals Let us consider an interval which the limits of the integral are themselves functions of α ∈ IR, it follows that: