  The quantum cohomology algebra of the (full) flag manifold is a fundamental
example in quantum cohomology theory, with connections to combinatorics,
algebraic geometry, and integrable systems. Using a differential geometric
approach, we give an algorithm for computing the multiplicative structure
constants of this algebra (the 3-point genus zero Gromov-Witten invariants).
The algorithm involves a Grobner basis calculation and the solution by
quadrature of a system of differential equations. In particular we obtain
quantum Schubert polynomials in a natural fashion.
