  We study convergence of the following discrete-time non-linear dynamical
system: n agents are located in R^d and at every time step, each moves
synchronously to the average location of all agents within a unit distance of
it. This popularly studied system was introduced by Krause to model the
dynamics of opinion formation and is often referred to as the Hegselmann-Krause
model. We prove the first polynomial time bound for the convergence of this
system in arbitrary dimensions. This improves on the bound of n^{O(n)}
resulting from a more general theorem of Chazelle. Also, we show a quadratic
lower bound and improve the upper bound for one-dimensional systems to O(n^3).
