# Incorrect Hessian when using \`exp\` function with complex input

**URL:** <https://discourse.julialang.org/t/incorrect-hessian-when-using-exp-function-with-complex-input/107739>\
**Category:** General Usage\
**Tags:** forwarddiff\
**Created:** [December 17, 2023, 4:57pm UTC](https://discourse.julialang.org/t/incorrect-hessian-when-using-exp-function-with-complex-input/107739 "2023-12-17T16:57:47Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![rz.yu](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/rz.yu/32/205484_2.png) [@rz.yu](https://discourse.julialang.org/u/rz.yu)\
**Post date:** [December 17, 2023, 4:57pm UTC](https://discourse.julialang.org/t/incorrect-hessian-when-using-exp-function-with-complex-input/107739/1 "2023-12-17T16:57:47Z")

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[Incorrect Hessian by `exp` function · Issue #677 · JuliaDiff/ForwardDiff.jl (github.com)](https://github.com/JuliaDiff/ForwardDiff.jl/issues/677)  
I was trying to calculate the Hessian of a simple function such as

```julia
using ForwardDiff
function busPQ(Ybus, Vθ)
    n = length(Vθ)
    V=Vθ[1:Int(n/2)]
    θ=Vθ[Int(n/2)+1:n]
    Vbus=V.*exp.(im .*θ)
    # Vbus=V.*(cos.(θ)+im .*sin.(θ))
    # print(Vbus-Vbus)
    I = Ybus.*Vbus
    return vcat(reim(Vbus.*conj(I))...)
end

Ybus=[1 - 3im]
Vm=[1]
Va=[0]
Vθ = vcat(Vm,Va)
function Hessian(f, x)
    n = length(x)
    out = ForwardDiff.jacobian(x -> ForwardDiff.jacobian(f, x), x)
    return reshape(out, n, n, n)
end
H(y) = Hessian(x-> busPQ(Ybus, x), y)

```

At point `[1,0]`, the above codes produce the wrong Hessian. However, if I set very small perturbation to `Vθ`, then the results are correct.

```julia
julia> H([1,0])
2×2×2 Array{Float64, 3}:
[:, :, 1] =
 2.0 0.0
 6.0 0.0

[:, :, 2] =
 0.0 2.0
 0.0 6.0

julia> H([1,1e-10])
2×2×2 Array{Float64, 3}:
[:, :, 1] =
 2.0 0.0
 6.0 0.0

[:, :, 2] =
 0.0 0.0
 0.0 0.0

julia> H([1,-1e-10])
2×2×2 Array{Float64, 3}:
[:, :, 1] =
 2.0 0.0
 6.0 0.0

[:, :, 2] =
 0.0 0.0
 0.0 0.0

```

It is found to be more robust to use `cos.(θ)+im .*sin.(θ)`, which functioned correctly at point `[1,0]`, instead of `exp.(im .*θ)`.  
I think the problem has something in common with #653.

The difference between `cos.(θ)+im .*sin.(θ)` and `exp.(im .*θ)` is printed as follows

```julia
Complex{ForwardDiff.Dual{ForwardDiff.Tag{var"#21#22", ForwardDiff.Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}, Int64, 2}}, ForwardDiff.Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}, Float64, 2}, 2}}[Dual{ForwardDiff.Tag{var"#21#22", ForwardDiff.Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}, Int64, 2}}}(Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}}(0.0,0.0,0.0),Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}}(0.0,0.0,0.0),Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}}(0.0,0.0,1.0)) + Dual{ForwardDiff.Tag{var"#21#22", ForwardDiff.Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}, Int64, 2}}}(Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}}(0.0,0.0,0.0),Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}}(0.0,0.0,0.0),Dual{ForwardDiff.Tag{var"#19#20"{var"#21#22"}, Int64}}(0.0,0.0,0.0))*im]2×2×2 Array{Float64, 3}:

```

It is clear that the above Dual is not all zero.

The dependencies are `ForwardDiff v0.10.36` and `Julia V1.9.3`.

---

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**Author:** ![mcabbott](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/mcabbott/32/6603_2.png) [@mcabbott](https://discourse.julialang.org/u/mcabbott)\
**Post date:** [December 17, 2023, 8:23pm UTC](https://discourse.julialang.org/t/incorrect-hessian-when-using-exp-function-with-complex-input/107739/2 "2023-12-17T20:23:32Z")

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Xref [issue 677](https://github.com/JuliaDiff/ForwardDiff.jl/issues/677), this appears to be solved on the unreleased v0.11.