  The newly developed Gorkov-Green's function approach represents a promising
path to the ab initio description of medium-mass open-shell nuclei. We discuss
the implementation of the method at second order with a two-body interaction,
with particular attention to the numerical solution of Gorkov's equation.
Different sources of theoretical error and degrees of self-consistency are
investigated. We show that Krylov projection techniques with a multi-pivot
Lanczos algorithm efficiently handle the growth of poles in the one-body
Green's function when Gorkov's equation is solved self-consistently. The end
result is a tractable, accurate and gently scaling ab initio scheme applicable
to full isotopic chains in the medium-mass region.
