Abstract. The rank-revealing URV decomposition is a useful tool for the subspace tracking problem in digital signal processing. Updating the decomposition is a stable process. However, downdating a rank-revealing URV decomposition could be unstable because the R factor is ill-conditioned. In this paper, we review some existing downdating algorithms for the full-rank URV decomposition in the absence of U and develop a new combined algorithm. We also show that the combined algorithm has relational stability. For the rank-revealing URV decomposition, we review a two-step method that applies full-rank downdating algorithms to the signal and noise parts separately. We compare several combinations of the full-rank algorithms and demonstrate good performance of our combined algorithm.