  We present a method to characterize and compute the large N formal
asymptotics of regular and critical Hermitian matrix models with general even
potentials in the one-cut and two-cut cases. Our analysis is based on a method
to solve continuum limits of the discrete string equation which uses the
resolvent of the Lax operator of the underlying Toda hierarchy. This method
also leads to an explicit formulation, in terms of coupling constants and
critical parameters, of the members of the Painlev\'e I and Painlev\'e II
hierarchies associated with one-cut and two-cut critical models respectively.
