  We provide an alternative Fourier analysis for multigrid applied to the
Poisson problem in 1D, based on explicit derivation of spectra of the iteration
matrix. The new Fourier analysis has advantages over the existing one. It is
easy to understand and enables us to write the error equation in terms of the
eigenvector of the stiffness matrix. When weighted-Jacobi is used as a smoother
with two different weights, multigrid is an exact solver.
