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 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 × 2 matrix representation thereof.

已知的复阶微分 (CSD) 方法通过使用紧邻实数轴的小虚阶来评估实解析函数，从而可以轻松且准确地将实解析函数微分到机器精度。当前的论文提出，可以通过在四元数方向上迈出一小步，以类似的方式计算全纯函数的导数。结果表明，这样做时，高精度和收敛性的 CSD 特性被转移到全纯函数的导数上。为了证明易于实现，使用复四元数、空间几何代数及其 2 × 2 矩阵表示进行了数值实验。