  We present a method for efficiently finding solutions of L\"uscher's
quantisation condition, the equation which relates two-particle scattering
amplitudes to the discrete spectrum of states in a periodic spatial volume of
finite extent such as that present in lattice QCD. The approach proposed is
based on an eigenvalue decomposition in the space of coupled-channels and
partial-waves, which proves to have several desirable and simplifying features
that are of great benefit when considering problems beyond simple elastic
scattering of spinless particles. We illustrate the method with a toy model of
vector-vector scattering featuring a high density of solutions, and with an
application to explicit lattice QCD energy level data describing $J^P=1^-$ and
$1^+$ scattering in several coupled channels.
