Complex-step-compatible atan2()

atan2() or arctan2() function that is compatible with complex-step derivative operations.

The function atan2(Y,X) [or arctan2(Y,X)] is the four quadrant arctangent of the elements of X and Y such that -pi <= atan2(Y,X) <= pi.

MATLAB atan2() reports an error when inputs are complex. This function reproduces MATLAB atan2() results for real inputs, but can be used with complex inputs to enable complex-step derivative operations.

There is an output discontinuity at T = pi and -pi. However, the output derivative is continuous, so, at T = pi and -pi, this function calculates the derivative analytically and adds complex-step-equivalent perturbations to the output.

Because the function remains discontinuous, the user should carefully consider before using atan2() output for gradient-based optimization.

Complex step is a numerical differentiation technique. See description:

https://blogs.mathworks.com/cleve/2013/10/14/complex-step-differentiation/