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A Guide to the Complex-Step Derivative Approximation

The complex-step is a method for estimating derivatives. This is a guide to the theory and implementation of the complex-step derivative approximation.

Introduction

The complex-step derivative approximation is a very convenient way of estimating derivatives numerically. It is a simple and accurate means of finding derivatives of a quantity calculated by an existing algorithm.

If $f$ is the result of any (real-valued) numerical algorithm, and $x$ is one of the input variables, then the derivative $\partial f / \partial x$ is simply 

$$ \frac{\partial f}{\partial x} \approx \frac{\Im \left[ f(x + ih) \right]}{ h } $$ 

where $i$ is the imaginary unit, and $h$ is a small step size. How small? h = 1.0E-20 works very well in nearly all cases, and results in derivatives no less accurate than the original algorithm.

The complex-step derivative approximation is a second order accurate formula analogous to finite-differencing, but it does not have a subtraction operation and therefore it is not subject to subtractive cancellation. Further details are available in the materials referenced below. 

Implementation

The implementation details of the complex-step derivative approximation are different for each programming language.

A noteworthy alternative to the complex-step method is the dual numbers approach. Please see this page for the implementation details. 

References