  For each integer $N\geq 2$, Mari\~no and Moore defined generalized Donaldson
invariants by the methods of quantum field theory, and made predictions about
the values of these invariants. Subsequently, Kronheimer gave a rigorous
definition of generalized Donaldson invariants using the moduli spaces of
anti-self-dual connections on hermitian vector bundles of rank $N$. In this
paper, Mari\~no and Moore's predictions are confirmed for simply connected
elliptic surfaces without multiple fibers and certain surfaces of general type
in the case that $N=3$. The primary motivation is to study 3-manifold instanton
Floer homologies which are defined by higher rank bundles. In particular, the
computation of the generalized Donaldson invariants are exploited to define a
Floer homology theory for sutured 3-manifolds.
