Membership test of ExponentialMap is incorrect

[schillic]

Example:

julia> n = 3;
julia> x = ones(n)
3-element Array{Float64,1}:
 1.0
 1.0
 1.0
julia> b = BallInf(ones(n), 0.1);

julia> sme = SparseMatrixExp(speye(n))
3×3 LazySets.SparseMatrixExp{Float64}:
 2.71828  0.0      0.0    
 0.0      2.71828  0.0    
 0.0      0.0      2.71828
julia> ex = sme * x
3-element Array{Float64,1}:
 2.71828
 2.71828
 2.71828
julia> em = sme * b;

julia> x ∈ b
true

Now since x ∈ b, we should expect that sme * x ∈ sme * b, or in short ex ∈ em.
However:

julia> sme * x ∈ sme * b
false
julia> ex ∈ em
false

[mforets]

True. The current exp.(A) is an element-wise exponentiation, while we really want expm there, or since we have a SparseMatrixExp, it is more appropriate to do

return ∈(expmv(-one(N), em.spmexp.M, x), em.X) 

[schillic]

The following doctest now does not work anymore.

julia> em = ExponentialMap(SparseMatrixExp(SparseMatrixCSC([2.0 0.0; 0.0 1.0])),
                           BallInf([1., 1.], 1.));

julia> ∈([5.0, 1.0], em)
false

Is the expected outcome correct?

[mforets]

nope. one can observe by plotting both em and Singleton(x) where x=[5, 1.] that x is in em.

[schillic]

How did you plot em? I get an error.

julia> LazySets.Approximations.box_approximation(em)
ERROR: AssertionError: radius must not be negative

Sounds like yet another bug 😞

[mforets]

julia> using LazySets, Plots

julia> em = ExponentialMap(SparseMatrixExp(SparseMatrixCSC([2.0 0.0; 0.0 1.0])),
                                  BallInf([1., 1.], 1.));

julia> s = Singleton([5.0, 1.0])
LazySets.Singleton{Float64}([5.0, 1.0])

julia> plot(em, alpha=0.5); plot!(s, alpha=0.4)

worked for me. also

julia> LazySets.Approximations.box_approximation(em)
LazySets.Hyperrectangle{Float64}([7.38906, 2.71828], [7.38906, 2.71828])

[schillic]

What the... it is nondeterministic! LazySets.Approximations.box_approximation(em) sometimes works and sometimes does not.
I created #486.