Hi all,

I am trying to differentiate a vector function with respect to a subset of the variables symbolically, but I am running into problems. Below is the problematic code:

```
using Symbolics, SparseArrays, LinearAlgebra, ForwardDiff
# setup data
θ = rand(3)
x = rand(11)
λ = rand(24)
S = sprand(Float64, 6, 7, 0.1)
Cp = sprand(Float64, 1, 11, 0.4)
_h = rand(24)
_d = spzeros(10)
λ = zeros(24)
θ = [30.0, 70.0, 90.0]
x = [1.9873817034700316, 3.9747634069400632, 1.9873817034700316, 1.9873817034700316, 1.9873817034700316, 3.9747634069400632, 1.9873817034700316, 0.028391167192429023, 0.022082018927444796, 0.13249211356466878, 0.13249211356466878]
λ[end] = -1.9873817034700316
S = [
1.0 0.0 0.0 0.0 0.0 0.0 -1.0
0.0 1.0 0.0 0.0 0.0 0.0 -2.0
0.0 0.0 1.0 0.0 0.0 0.0 -1.0
0.0 0.0 0.0 1.0 0.0 0.0 -1.0
0.0 0.0 0.0 0.0 1.0 0.0 -1.0
0.0 0.0 0.0 0.0 0.0 1.0 -2.0
]
se_vars = ((2, -1.0), (3, -1.0),(3, -3.0), (1, -2.0), (1, -1.0),)
# create functions
Se(θ) = sparse(
[1, 2, 3, 4, 3],
[3, 4, 4, 5, 5],
[k / θ[idx] for (idx, k) in se_vars],
4,
7,
)
E(θ) = [
S spzeros(6, 4)
Se(θ) I(4)
]
d = _ -> _d
M = _ -> [
-I(11)
I(11)
-Cp
Cp
]
h = _ -> _h
# make symbolic variables
Symbolics.@variables begin
sx[1:11]
sλ[1:24]
sθ[1:3]
end
# define function to differentiate
sparse_F(x, λ, θ) = [
E(θ) * x - d(θ)
λ .* (M(θ) * x - h(θ))
]
sj = Symbolics.jacobian(sparse_F(sx, sλ, sθ), collect(sθ)) # jacobian wrt θ only (collecting comes from the linked to other issue on discourse)
# this yields a matrix of all zeros
# check the answer with ForwardDiff
dense_F(x, λ, θ) = [
Array(E(θ)) * x - Array(d(θ))
λ .* (Array(M(θ)) * x - Array(h(θ)))
]
j = ForwardDiff.jacobian(θ -> dense_F(x, λ, θ), θ) # this is different :(
```

Weirdly, if I don’t write the function out nicely (as above), but group all the variables together, symbolics correctly calculates the Jacobian. From here I can just extract the part I care about. However, for the application I am working on this is a huge matrix., and evaluating the Jacobian this way causes Julia to crash… Here is the code that works, but calculates all the derivatives and crashes for big systems:

```
xidxs = 1:11
λidxs = last(xidxs) .+ (1:24)
θidxs = last(λidxs) .+ (1:3)
sparse_F2(z) = [
E(z[θidxs]) * z[xidxs] - d(z[θidxs])
spdiagm(z[λidxs]) * (M(z[θidxs]) * z[xidxs] - h(z[θidxs]))
]
sz = [sx; sλ; sθ]
sparse_F2(sz)
sj = Symbolics.jacobian(sparse_F2(sz), sz)[:, end-3:end] # need to separate it out
```

Does anybody have any ideas on how to make the first code snippet work? I think this may also be related to this problem.