We provide an algorithm to compute an effective description of the homology
of complex projective hypersurfaces relying on Picard-Lefschetz theory. Next,
we use this description to compute high-precision numerical approximations of
the periods of the hypersurface. This is an improvement over existing
algorithms as this new method allows for the computation of periods of smooth
quartic surfaces in an hour on a laptop, which could not be attained with
previous methods. The general theory presented in this paper can be generalised
to varieties other than just hypersurfaces, such as elliptic fibrations as
showcased on an example coming from Feynman graphs. Our algorithm comes with a
SageMath implementation.Comment: 35 page