  We present a new numerical method for computing the pure-point spectrum
associated with the linear stability of coherent structures. In the context of
the Evans function shooting and matching approach, all the relevant information
is carried by the flow projected onto the underlying Grassmann manifold. We
show how to numerically construct this projected flow in a stable and robust
manner. In particular, the method avoids representation singularities by, in
practice, choosing the best coordinate patch representation for the flow as it
evolves. The method is analytic in the spectral parameter and of complexity
bounded by the order of the spectral problem cubed. For large systems it
represents a competitive method to those recently developed that are based on
continuous orthogonalization. We demonstrate this by comparing the two methods
in three applications: Boussinesq solitary waves, autocatalytic travelling
waves and Ekman boundary layer.
