  We give an algorithm to compute the periods of smooth projective
hypersurfaces of any dimension. This is an improvement over existing algorithms
which could only compute the periods of plane curves. Our algorithm reduces the
evaluation of period integrals to an initial value problem for ordinary
differential equations of Picard-Fuchs type. In this way, the periods can be
computed to extreme-precision in order to study their arithmetic properties.
The initial conditions are obtained by an exact determination of the cohomology
pairing on Fermat hypersurfaces with respect to a natural basis.
