Computing periods without taking Riemann integrals

We will describe an algorithm to compute the period matrix of projective hypersurfaces, in particular, of plane curves. This algorithm computes the value of the period integrals without taking any Riemann integrals but by solving an initial value problem for a system of ordinary differential equations. With this approach we bypass the need to determine explicit homology cycles on the hypersurfaces, which is the key to period computation beyond dimension one. Furthermore, as quadrature is avoided, periods can be computed to extreme-precision.