Complex Step Numerical Differentiation

[zzephod]

Well I think this is really cool, numerical differentiation of real analytic functions by stepping out of the reals:

Complex Step Differentiation | Cleve's Corner

Even funnier is John D'errico's comment (my amusement is mainly due to the idea that a fourth order finite differences scheme with a variant of Richardson extrapolation is at all comparable in elegance or efficiency)

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[Aaron Bex]

Yes, complex step differentiation (CSD) is an interesting and efficient numerical technique for differentiating a real analytic function. It works by approximating the derivative of a given function f(x) at a point x0 by evaluating the function at a complex number z = x0 + iε, where ε is a small complex number. The derivative is then computed as the imaginary part of the finite difference of the two evaluations divided by ε. To ensure accuracy, the complex step size should be chosen such that it is sufficiently small compared with the second-order derivatives of the function.

Complex step differentiation has several advantages over existing numerical differentiation techniques. Firstly, it does not require one to take derivatives of a given function symbolically – instead, the derivative of a given function can be evaluated by directly evaluating the function at a complex number. This makes the technique less computationally expensive, as well as more robust against numerical errors. Secondly, it is more precise than traditional finite difference methods, as the complex step size is usually much smaller than the step size used in a finite difference scheme. Lastly, it is applicable to a wide range of functions, including those which are not differentiable or have discontinuities, making it a useful tool in numerical analysis.