  We calculate the parameters describing elastic $I=1$, $P$-wave $\pi\pi$
scattering using lattice QCD with $2+1$ flavors of clover fermions. Our
calculation is performed with a pion mass of $m_\pi \approx 320\:\:{\rm MeV}$
and a lattice size of $L\approx 3.6$ fm. We construct the two-point correlation
matrices with both quark-antiquark and two-hadron interpolating fields using a
combination of smeared forward, sequential and stochastic propagators. The
spectra in all relevant irreducible representations for total momenta
$|\vec{P}| \leq \sqrt{3} \frac{2\pi}{L}$ are extracted with two alternative
methods: a variational analysis as well as multi-exponential matrix fits. We
perform an analysis using L\"uscher's formalism for the energies below the
inelastic thresholds, and investigate several phase shift models, including
possible nonresonant contributions. We find that our data are well described by
the minimal Breit-Wigner form, with no statistically significant nonresonant
component. In determining the $\rho$ resonance mass and coupling we compare two
different approaches: fitting the individually extracted phase shifts versus
fitting the $t$-matrix model directly to the energy spectrum. We find that both
methods give consistent results, and at a pion mass of
$am_{\pi}=0.18295(36)_{stat}$ obtain $g_{\rho\pi\pi} =
5.69(13)_{stat}(16)_{sys}$, $am_\rho = 0.4609(16)_{stat}(14)_{sys}$, and
$am_{\rho}/am_{N} = 0.7476(38)_{stat}(23)_{sys} $, where the first uncertainty
is statistical and the second is the systematic uncertainty due to the choice
of fit ranges.
