  The $(x, y)$ position reconstruction method used in the analysis of the
complete exposure of the Large Underground Xenon (LUX) experiment is presented.
The algorithm is based on a statistical test that makes use of an iterative
method to recover the photomultiplier tube (PMT) light response directly from
the calibration data. The light response functions make use of a two
dimensional functional form to account for the photons reflected on the inner
walls of the detector. To increase the resolution for small pulses, a photon
counting technique was employed to describe the response of the PMTs. The
reconstruction was assessed with calibration data including
${}^{\mathrm{83m}}$Kr (releasing a total energy of 41.5 keV) and ${}^{3}$H
($\beta^-$ with Q = 18.6 keV) decays, and a deuterium-deuterium (D-D) neutron
beam (2.45 MeV). In the horizontal plane, the reconstruction has achieved an
$(x, y)$ position uncertainty of $\sigma$= 0.82 cm for events of only 200
electroluminescence photons and $\sigma$ = 0.17 cm for 4,000
electroluminescence photons. Such signals are associated with electron recoils
of energies $\sim$0.25 keV and $\sim$10 keV, respectively. The reconstructed
position of the smallest events with a single electron emitted from the liquid
surface has a horizontal $(x, y)$ uncertainty of 2.13 cm.
