  We report on calculations of the charmonium and bottomonium spectrum in
lattice QCD. We use ensembles of gauge fields with three flavors of sea quarks,
simulated with the asqtad improved action for staggered fermions. For the heavy
quarks we employ the Fermilab interpretation of the clover action for Wilson
fermions. These calculations provide a test of lattice QCD, including the
theory of discretization errors for heavy quarks. We provide, therefore, a
careful discussion of the results in light of the heavy-quark effective
Lagrangian. By and large, we find that the computed results are in agreement
with experiment, once parametric and discretization errors are taken into
account.
