We develop numerical homotopy algorithms for solving systems of
polynomial equations arising from the classical Schubert calculus. These
homotopies are optimal in that generically no paths diverge. For systems
obtained from hypersurface Schubert conditions we give two algorithms based
on extrinsic deformations of the Grassmannian: one is derived from a
Gröbner basis for the Plücker ideal of the Grassmannian and the
other from a SAGBI basis for its projective coordinate ring. The more
general case of special Schubert conditions is solved by delicate intrinsic
deformations, called Pieri homotopies, which first arose in the study of
enumerative geometry over the real numbers. Computational results are
presented and applications to control theory are discussed.