  An enriched approximation space is the span of a conventional basis with a
few extra functions included, for example to capture known features of the
solution to a computational problem. Adding functions to a basis makes it
overcomplete and, consequently, the corresponding discretized approximation
problem may require solving an ill-conditioned system. Recent research
indicates that these systems can still provide highly accurate numerical
approximations under reasonable conditions. In this paper we propose an
efficient algorithm to compute such approximations. It is based on the AZ
algorithm for overcomplete sets and frames, which simplifies in the case of an
enriched basis. In addition, analysis of the original AZ algorithm and of the
proposed variant gives constructive insights on how to achieve optimal and
stable discretizations using enriched bases. We apply the algorithm to examples
of enriched approximation spaces in literature, including a few non-standard
approximation problems and an enriched spectral method for a 2D boundary value
problem, and show that the simplified AZ algorithm is indeed stable, accurate
and efficient.
