Root finding fails for functions with complex numbers

[dpsanders]

julia> g(x) = x^3 + x^2 + 3x + im

julia> roots(g, Complex(-5..5, -5..5))
promotion of types IntervalRootFinding.Compl{Interval{Float64}} and Complex{Int64} failed to change any arguments

[dpsanders]

julia> IntervalRootFinding.roots(z->z^3 + z^2 + 3z + 1, Complex(-5..5, -5..5))
TypeError: non-boolean (Interval{Float64}) used in boolean context

[hongchengni]

Support. It would be great if complex root finding is supported.

[dpsanders]

Since we can extract the Jacobian matrix of the pair of real functions (realpart, imagpart) from the complex derivative, it should be enough to provide the complex derivative, or calculate that using automatic differentiation if possible.

[dpsanders]

From Slack (Mason Protter): AD for complex functions works using ForwardDiff2.jl:

julia> using ForwardDiff2: DI

julia> f(z) = z^2 / (2z + 1)
f (generic function with 1 method)

julia> z = 1 + im
1 + 1im

julia> DI(f)(z) ≈ 2z*(z+1) / (2z + 1)^2
true