Numerically stable computation of embedding formulae for scattering by polygons

For problems of time-harmonic scattering by polygonal obstacles, embedding formulae provide a useful means of computing the far-field coefficient induced by any incident plane wave, given the far-field coefficient of a relatively small set of canonical problems. The number of such problems to be solved depends only on the geometry of the scatterer. Whilst the formulae themselves are exact in theory, any implementation will inherit numerical error from the method used to solve the canonical problems. This error can lead to numerical instabilities. Here, we present an effective approach to identify and regulate these instabilities. This approach is subsequently extended to the case where the incident wave is a Herglotz wave function, and we suggest how this could potentially remove frequency dependence of a T-matrix method.