Quaternionic step derivative: Machine precision differentiation of holomorphic functions using complex quaternions

The known Complex Step Derivative (CSD) method allows easy and accurate differentiation up to machine precision of real analytic functions by evaluating them with a small imaginary step next the number line. current paper proposes that derivatives holomorphic can be calculated in similar fashion taking quaternionic direction instead. It is demonstrated so doing CSD properties high accuracy convergence are carried over functions. To demonstrate ease implementation, numerical experiments were performed using complex quaternions, geometric algebra space, 2 × matrix representation thereof.