Iteratively compute eigenvalues of non-Hermitian sparse matrix

Let A be a real, non-symmetric, sparse matrix with eigenvalues having real part smaller than or equal to 0. We can assume that if the real part of an eigenvalue λ is 0 then λ=0. The multiplicity of the 0 eigenvalue is greater than or equal to 1.

I want to iteratively calculate the eigenvalues with largest real part (beginning from 0) and stop, whenever an eigenvalue with real part < 0 was found. I want to record that eigenvalue. I do not care about the eigenvectors.

With what Julia package can I do that? I am especially interested in aborting the calculation of largest real part eigenvalues, when an eigenvalue with real part < 0 was found.