  The Variational Theory of Complex Rays (VTCR) is an indirect Trefftz method
designed to study systems governed by Helmholtz-like equations. It uses wave
functions to represent the solution inside elements, which reduces the
dispersion error compared to classical polynomial approaches but the resulting
system is prone to be ill conditioned. This paper gives a simple and original
presentation of the VTCR using the discontinuous Galerkin framework and it
traces back the ill-conditioning to the accumulation of eigenvalues near zero
for the formulation written in terms of wave amplitude. The core of this paper
presents an efficient solving strategy that overcomes this issue. The key
element is the construction of a search subspace where the condition number is
controlled at the cost of a limited decrease of attainable precision. An
augmented LSQR solver is then proposed to solve efficiently and accurately the
complete system. The approach is successfully applied to different examples.
