Complex-valued arguments

[baggepinnen]

Currently, the implicit function is defined for real arguments only

function (implicit::ImplicitFunction)(
    x_and_dx::AbstractArray{Dual{T,R,N}}; kwargs...
) where {T,R,N}

It would be useful to have it work for complex-valued arguments as well. Naively defining the same function with modified type signature did not seem to work.

[gdalle]

I have no clue how complex differentiation works, does the implicit function theorem apply too?

[gdalle]

Quote from @baggepinnen:

The problem is that the duals are placed inside the Complex type, i.e., Complex{Dual}, and handling might require overloading ForwardDiff.value so that
x = value.(x_and_dx)
works as expected, and the same for partials etc. I tried very briefly but didn't get it to work, so it probably requires some digging to figure out.