  We study the change of moduli spaces of Gieseker-semistable torsion free
rank-$2$ sheaves on algebraic surfaces as we vary the polarizations. When the
surfaces are rational with an effective anti-canonical divisor, the moduli
spaces are linked by a series of flips (blowups and blowdowns). Using these
results, we compute the transition formulas for Donaldson polynomial invariants
of rational surfaces. Part of the work is also obtained independently by
Matsuki-Wentworth and Ellingsrud-G{\" o}ttsche.
