Add support for complex numbers

[zoharl3]

Something like:

VectorXcd b;
b << 1, 2;
typedef Matrix<forward::Dual<complex<double>, complex<double>>, -1, 1, 0> VectorXcdual;
VectorXcdual f;
auto r = b.transpose() * f;

[allanleal]

Hi @zoharl3 - sorry for the delay.

Have you tried instead using std::complex<dual>?

[zoharl3]

Looks like it's working. Not sure if perfectly, e.g. I can assign (or multiply) VectorXd to (by) VectorXdual, but not VectorXcd to (by) VectorXcdual. Maybe it needs casting or something; but I'll look into it more closely when I need it. For now, I just used real implementation.

If you think that it works smoothly, then you may want to consider adding an example to the tutorial, which others may find useful.

[petar-andrejic]

It would be nice if rather than being distributed across a few tutorials and GitHub issues, that the complex support was folded into the library itself. As it stands I'm having a hard time getting it working properly. For example, using std::complex refuses to compile if the code uses derivative(...), while using Dual<std::complex, std::complex> like the tutorial suggests is also somewhat problematic. For example, it doesn't have real() or imag() methods like std::complex does, and things like Eigen want it to. Furthermore, some things seem to be hardcoded in a problematic way, for example, while almost every operator is made generic using UnaryExpr, for conj, real and imag the methods are hardcoded to return the number itself, and 0 respectively i.e. the Dual class will only work correctly for real types. As such, it seems that there need to be some fixes internally before complex numbers work properly

[allanleal]

Hi @petar-andrejic , thanks for your investigation. As discussed in #297, if you work on that Pull Request, I'll be happy to merge.