  We match three hadronic equations of state at low energy densities to a
perturbatively computed equation of state of quarks and gluons at high energy
densities. One of them includes all known hadrons treated as point particles,
which approximates attractive interactions among hadrons. The other two
include, in addition, repulsive interactions in the form of excluded volumes
occupied by the hadrons. A switching function is employed to make the crossover
transition from one phase to another without introducing a thermodynamic phase
transition. A chi-square fit to accurate lattice calculations with temperature
$100 < T < 1000$ MeV determines the parameters. These parameters quantify the
behavior of the QCD running gauge coupling and the hard core radius of protons
and neutrons, which turns out to be $0.62 \pm 0.04$ fm. The most physically
reasonable models include the excluded volume effect. Not only do they include
the effects of attractive and repulsive interactions among hadrons, but they
also achieve better agreement with lattice QCD calculations of the equation of
state. The equations of state constructed in this paper do not result in a
phase transition, at least not for the temperatures and baryon chemical
potentials investigated. It remains to be seen how well these equations of
state will represent experimental data on high energy heavy ion collisions when
implemented in hydrodynamic simulations.
