In "Some numerical results in complex differential geometry," Donaldson defines a dynamical system on the space of Fubini-Study metrics
on a polarized compact Kähler manifold. Sano proved that if there exists a balanced metric for the
polarization, then this dynamical system always converges to the balanced metric (Y. Sano, "Numerical algorithm for finding balanced metrics").
In "Numerical solution to the Hermitian Yang-Mills equation on the Fermat quintic," Douglas, et. al., conjecture that the same holds in the case
of vector bundles. In this paper, we give an affirmative answer to their conjecture.