The complex-step derivative approximation

TLDR

The paper introduces the complex‑step derivative approximation, proposes accuracy‑enhancing improvements, and connects it to algorithmic differentiation theory. The authors detail a general implementation procedure, provide an automation script, deliver automatic Fortran and C/C++ implementations, and evaluate the method in two large multidisciplinary solvers against finite‑difference and existing AD tools. The complex‑step method yields sensitivities as accurate as finite‑difference results while offering superior accuracy, robustness, ease of implementation, and maintainability, making it an attractive option for sensitivity analysis.

Abstract

The complex-step derivative approximation and its application to numerical algorithms are presented. Improvements to the basic method are suggested that further increase its accuracy and robustness and unveil the connection to algorithmic differentiation theory. A general procedure for the implementation of the complex-step method is described in detail and a script is developed that automates its implementation. Automatic implementations of the complex-step method for Fortran and C/C++ are presented and compared to existing algorithmic differentiation tools. The complex-step method is tested in two large multidisciplinary solvers and the resulting sensitivities are compared to results given by finite differences. The resulting sensitivities are shown to be as accurate as the analyses. Accuracy, robustness, ease of implementation and maintainability make these complex-step derivative approximation tools very attractive options for sensitivity analysis.