  We report on an all-sky search for periodic gravitational waves in the
frequency range $\mathrm{50-1000 Hz}$ with the first derivative of frequency in
the range $-8.9 \times 10^{-10}$ Hz/s to zero in two years of data collected
during LIGO's fifth science run. Our results employ a Hough transform
technique, introducing a $\chi^2$ test and analysis of coincidences between the
signal levels in years 1 and 2 of observations that offers a significant
improvement in the product of strain sensitivity with compute cycles per data
sample compared to previously published searches. Since our search yields no
surviving candidates, we present results taking the form of frequency
dependent, 95$%$ confidence upper limits on the strain amplitude $h_0$. The
most stringent upper limit from year 1 is $1.0\times 10^{-24}$ in the
$\mathrm{158.00-158.25 Hz}$ band. In year 2, the most stringent upper limit is
$\mathrm{8.9\times10^{-25}}$ in the $\mathrm{146.50-146.75 Hz}$ band. This
improved detection pipeline, which is computationally efficient by at least two
orders of magnitude better than our flagship Einstein$@$Home search, will be
important for "quick-look" searches in the Advanced LIGO and Virgo detector
era.

General Relativity and Quantum Cosmology