# Automatic Differentiation over Vectors

#1

Consider the following expression `c=exp(A*t)*b` where `A` is a matrix, `b` and `c` are vectors and `t` is a scalar. I’m trying to teach ForwardDiff to calculate the derivative of `c` with respect to `t`, which is nothing but `A*c`. This is my attempt so far

``````import ForwardDiff: derivative, Dual, value, partials
import ExponentialUtilities: expv
using SparseArrays

function expv(t::Dual, A, v::Vector; kw...)
w = expv(value(t), A, v; kw...)
return Dual(w, (A*w) .* partials(t))
end

A = sparse([0 1.; 1. 0])
v = [1., 0]

f(t) = expv(t, A, v)

derivative(f, 1.0)
``````

However, I get the following error, which makes me think that this is not possible at all. Is this the case? Is there any other automatic differentiation tool for doing this?

``````ERROR: ArgumentError: Cannot create a dual over scalar type Array{Float64,1}. If the type behaves
as a scalar, define FowardDiff.can_dual.
``````

#2

I haven’t tried this, but I believe `ForwardDiff.Dual` needs to be the elements of the array `w`. It may be as simple as `Dual.(w, …)`. (But it may also just work without any definitions?)

The various reverse-diff packages (Flux, Zygote, Yota) all have ways to define custom gradients, but the reverse one (I guess `dot(Δ,W*c)`?)