  This paper introduces the localized sparsifying preconditioner for the
pseudospectral approximations of indefinite systems on periodic structures. The
work is built on top of the recently proposed sparsifying preconditioner with
two major modifications. First, the local potential information is utilized to
improve the accuracy of the preconditioner. Second, an FFT based method to
compute the local stencil is proposed to reduce the setup time of the
algorithm. Numerical results show that the iteration number of this improved
method grows only mildly as the problem size grows, which implies that solving
pseudospectral approximation systems is computationally as efficient as solving
sparse systems, up to a mildly growing factor.
