  In this paper, we present two improved Gauss-Seidel projection methods with
unconditional stability. The first method updates the gyromagnetic term and the
damping term simultaneously and follows by a projection step. The second method
introduces two sets of approximate solutions, where we update the gyromagnetic
term and the damping term simultaneously for one set of approximate solutions
and apply the projection step to the other set of approximate solutions in an
alternating manner. Compared to the original Gauss-Seidel projection method
which has to solve heat equations $7$ times at each time step, the improved
methods solve heat equations $5$ times and $3$ times, respectively. First-order
accuracy in time and second-order accuracy in space are verified by examples in
both 1D and 3D. In addition, unconditional stability with respect to both the
grid size and the damping parameter is confirmed numerically. Application of
both methods to a realistic material is also presented with hysteresis loops
and magnetization profiles. Compared with the original method, the recorded
running times suggest that savings of both methods are about $2/7$ and $4/7$
for the same accuracy requirement, respectively.
