Fix error message for LDL factorization failures

[odow]

Found by solver tests: https://github.com/jump-dev/MathOptInterface.jl/actions/runs/22469911691/job/65084327414

[odow]

I'm not sure what's going on here. LDLFactorizations.factorized returns false, but there is a valid factorisation:

julia> import LDLFactorizations

julia> import LinearAlgebra

julia> Q = [1.0 1.0; 1.0 1.0]
2×2 Matrix{Float64}:
 1.0  1.0
 1.0  1.0

julia> f = LDLFactorizations.ldl(Q)
LDLFactorizations.LDLFactorization{Float64, Int64, Int64, Int64}(true, false, false, 2, [2, -1], [1, 0], [2, 2], [1, 2], [1, 2], [1, 2, 2], Int64[], Int64[], [2], [1.0], [1.0, 0.0], [0.0, 0.0], [1, 1], 0.0, 0.0, 0.0, 2)

julia> @assert f.P == [1, 2]

julia> L = Matrix(f.L) .+ LinearAlgebra.I(2)
2×2 Matrix{Float64}:
 1.0  0.0
 1.0  1.0

julia> D = Matrix(f.D)
2×2 Matrix{Float64}:
 1.0  0.0
 0.0  0.0

julia> LDLFactorizations.factorized(f)
false

julia> L * D * L'
2×2 Matrix{Float64}:
 1.0  1.0
 1.0  1.0

cc @amontoison any ideas?

[amontoison]

@odow I checked the code and we stop the factorization when we encounter D[k] == 0 and we return false for "factorized".

But if k == n, we reached the end of the factorization and we can return true.

[odow]

Is that intentional or a bug?

[amontoison]

It was intentional because we can't do ldiv! or \ if we have D[k] == 0.
But we can still do mul!.
I added another attribute to specify if we have a singularity of not.

[odow]

Merging because this is an improvement for now. We can update when a new version of LDLFactorizations is released