  We propose an iterative solution method for the 3D high-frequency Helmholtz
equation that exploits a contour integral formulation of spectral projectors.
In this framework, the solution in certain invariant subspaces is approximated
by solving complex-shifted linear systems, resulting in faster GMRES iterations
due to the restricted spectrum. The shifted systems are solved by exploiting a
polynomial fixed-point iteration, which is a robust scheme even if the
magnitude of the shift is small. Numerical tests in 3D indicate that
$O(n^{1/3})$ matrix-vector products are needed to solve a high-frequency
problem with a matrix size $n$ with high accuracy. The method has a small
storage requirement, can be applied to both dense and sparse linear systems,
and is highly parallelizable.
