  We introduce a new algorithm for computing the periods of a smooth complex
projective hypersurface. The algorithm intertwine with a new method for
computing an explicit basis of the singular homology of the hypersurface. It is
based on Picard-Lefschetz theory and relies on the computation of the monodromy
action induced by a one-parameter family of hyperplane sections on the homology
of a given section. We provide a SageMath implementation. For example, on a
laptop, it makes it possible to compute the periods of a smooth complex quartic
surface with hundreds of digits of precision in typically an hour.
