add oop jac forwarddiff_color_jacobian

[huanglangwen]

forwarddiff_color_jacobian(oop_f, x, sparsity = nothing) == ForwardDiff.jacobian(oop_f, x)
jacout::Array{T,2} = forwarddiff_color_jacobian(oop_f, x, colorvec = ... , sparsity = ... )
jacout::T = forwarddiff_color_jacobian(oop_f, x, colorvec = ... , sparsity = ... , jac_prototype :: T = jac)

[huanglangwen]

Is it good for merge? @ChrisRackauckas

[ChrisRackauckas]

we should make this test be on static arrays instead, so it tests that no mutation is used by failing on mutation

[ChrisRackauckas]

there should be a way to remove the array allocation here.

[huanglangwen]

I'm thinking of the delta tensor but that would introduce yet another dependency.

[ChrisRackauckas]

The delta tensor? Is that a FillArrays.jl object?

[huanglangwen]

I thought there would be some lazy Kronecker delta tensors, but looks like there's no such thing. FillArrays.jl doesn't work, the slice of eye generates sparse vector and its product with a normal array would be sparseCSC which is worse.

[huanglangwen]

I can't find any to make a general way of removing this array allocation. If we focus at StaticArrays, maybe I can make some overloads for + .

[ChrisRackauckas]

It would be fine to have a special branch for StaticArrays. How does ForwardDiff do it? It has a special routine for Jacobians that doesn't allocate.

[ChrisRackauckas]

To clarify the purpose here, the OOP version is really for two cases:

1. Static arrays. This is for very small systems and shouldn't allocate at all. This should be the fastest version of the function possible, so hopefully we the differentiation code won't slow it down.
2. Some forms of automatic differentiation do best with OOP code, and when they do so they will be allocating so we "don't need to care" about speed in this case.

So we really just need to make sure that SArray works and that case is fast.