  We show how direct integration can be used to solve the closed amplitudes of
multi-cut matrix models with polynomial potentials. In the case of the cubic
matrix model, we give explicit expressions for the ring of non-holomorphic
modular objects that are needed to express all closed matrix model amplitudes.
This allows us to integrate the holomorphic anomaly equation up to holomorphic
modular terms that we fix by the gap condition up to genus four. There is an
one-dimensional submanifold of the moduli space in which the spectral curve
becomes the Seiberg--Witten curve and the ring reduces to the non-holomorphic
modular ring of the group $\Gamma(2)$. On that submanifold, the gap conditions
completely fix the holomorphic ambiguity and the model can be solved explicitly
to very high genus. We use these results to make precision tests of the
connection between the large order behavior of the 1/N expansion and
non-perturbative effects due to instantons. Finally, we argue that a full
understanding of the large genus asymptotics in the multi-cut case requires a
new class of non-perturbative sectors in the matrix model.
