Fix type piracies

[nathanaelbosch]

Current output:

julia> Aqua.test_piracies(ProbNumDiffEq)
Possible type-piracy detected:
[1] *(M::AbstractMatrix, g::Gaussian{VM, PSDMatrix{T, S}} where {T, S, VM<:AbstractVecOrMat{T}}) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:25
[2] X_A_Xt(A, X) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/ProbNumDiffEq.jl:37
[3] copy(K::Kronecker.KroneckerProduct) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/kronecker.jl:1
[4] copy(P::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:4
[5] copy!(dst::Gaussian, src::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:6
[6] isapprox(M1::PSDMatrix, M2::PSDMatrix; kwargs...) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/filtering/markov_kernel.jl:48
[7] kwcall(::Any, ::typeof(isapprox), M1::PSDMatrix, M2::PSDMatrix) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/filtering/markov_kernel.jl:48
[8] mean(s::StructArray{Gaussian{VM, PSDMatrix{T, S}} where VM<:AbstractVecOrMat{T}} where {T, S}) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:35
[9] ndims(g::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:17
[10] recursivecopy(P::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:11
[11] recursivecopy!(dst::Gaussian, src::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:12
[12] show(io::IO, g::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:13
[13] similar(P::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:5
[14] size(g::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:16
[15] std(g::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:19
[16] var(g::Gaussian) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/gaussians.jl:18

[nathanaelbosch]

PSDMatrices.jl now has isapprox: nathanaelbosch/PSDMatrices.jl#31

[nathanaelbosch]

To get rid of more, we could move from GaussianDistributions.jl, which seems quite unmaintained, to the much bigger Distributions.jl. The main issue here is that MvNormal requires a covariance of type AbstractPDMat, so this would basically require PSDMatrices.jl to move towards being a subtype of AbstractPDMat. This could work!

Alternatively, we could also make our own small little Gaussian distribution package, like GaussianDistributions.jl but focusing only on the multivariate case, mostly considering PSDMatrices as covariances, witht speed in mind, and most importantly, maintained.

[nathanaelbosch]

Turns out Distributions.jl does not have copy!, similar, etc. So SimpleGaussians.jl it is?

[nathanaelbosch]

Current output:

julia> Aqua.test_piracies(ProbNumDiffEq)
Possible type-piracy detected:
[1] X_A_Xt(A, X) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/ProbNumDiffEq.jl:42
[2] copy(K::Kronecker.KroneckerProduct) @ ProbNumDiffEq ~/.julia/dev/ProbNumDiffEq/src/kronecker.jl:1