Quaternionic Step Derivative: Machine Precision Differentiation of Holomorphic Functions using Complex Quaternions

TL;DR

This paper extends the complex step derivative method to holomorphic functions using complex quaternions, enabling highly accurate derivative calculations with simple implementation and demonstrated through numerical experiments.

Contribution

It introduces a quaternionic step derivative method for holomorphic functions, preserving high accuracy and convergence properties of the original complex step derivative approach.

Findings

Achieves machine precision in derivative calculations for holomorphic functions.

Demonstrates the method's effectiveness through numerical experiments.

Provides a straightforward implementation using complex quaternions.

Abstract

The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them a small imaginary step next to the real number 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 CSD properties of high accuracy and convergence are carried over to derivatives of holomorphic functions. To demonstrate the ease of implementation, numerical experiments were performed using complex quaternions, the geometric algebra of space, and a $2 \times 2$ matrix representation thereof.