On Automatic Differentiation of Codes with COMPLEX Arithmetic with Respect to Real Variables

We explore what it means to apply automatic differentiation with respect to a set of real variables to codes containing complex arithmetic. That is, both dependent and independent variables with respect to differentiation are real variables, but in order to exploit features of complex mathematics, part of the code is expressed by employing complex arithmetic. We investigate how one can apply automatic differentiation to complex variables if one exploits the homomorphism of the complex numbers C onto R 2 . It turns out that, by and large, the usual rules of differentiation apply, but subtle differences in special cases arise for sqrt(), abs(), and the power operator. 1 Introduction In physics and engineering applications, while the underlying independent variables and operations are intrinsically real-valued, it is nevertheless often convenient to employ complex-valued representations. Applications of complex function theory in applied mathematics, physics, and engineering (and hence..