  A new deterministic, numerical method to solve fermion field theories is
presented. This approach is based on finding solutions $Z[J]$ to the lattice
functional equations for field theories in the presence of an external source
$J$. Using Grassmann polynomial expansions for the generating functional $Z$,
we calculate propagators for systems of interacting fermions. These
calculations are straightforward to perform and are executed rapidly compared
to Monte Carlo. The bulk of the computation involves a single matrix inversion.
Because it is not based on a statistical technique, it does not have many of
the difficulties often encountered when simulating fermions. Since no
determinant is ever calculated, solutions to problems with dynamical fermions
are handled more easily. This approach is very flexible, and can be taylored to
specific problems based on convenience and computational constraints. We
present simple examples to illustrate the method; more general schemes are
desirable for more complicated systems.
