  The limit n to infinity of the classical O(n) phi^4 model on a 3d film with
free surfaces is studied. Its exact solution involves a self-consistent 1d
Schr\"odinger equation, which is solved numerically for a partially discretized
as well as for a fully discrete lattice model. Numerically exact results are
obtained for the scaled Casimir force at all temperatures. Obtained via a
single framework, they exhibit all relevant qualitative features of the
thermodynamic Casimir force known from wetting experiments on Helium-4 and
Monte Carlo simulations, including a pronounced minimum below the bulk critical
point.
