Quaternionic Step Derivative: Automatic Differentiation of Holomorphic Functions using Complex Quaternions
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Complex Step Derivative (CSD) allows easy and accurate differentiation up to machine precision of real functions by evaluating them a small imaginary step next to the real line. The current paper proposes that derivatives of holomorphic functions can be calculated in a similar fashion by taking a small step in a quaternionic direction instead. It is demonstrated that in so doing the properties of high accuracy and convergence are carried over to derivatives of holomorphic functions. Additionally, the extra degree of freedom means second derivatives of real functions are straightforwardly computed by taking a small step in both a quaternionic and complex direction, making the technique a powerful extension of CSD. To demonstrate the ease of implementation numerical experiments were performed using complex quaternions, the geometric algebra of space, and a 2 × 2 matrix representation thereof.
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