The authors propose a new approach for the adaptive identification of sparse systems. This approach improves on the recursive least squares (RLS) algorithm by adding a sparsity inducing weighted l(1) norm penalty to the RLS cost function. Subgradient analysis is utilised to develop the recursive update equations for the calculation of the optimum system estimate, which minimises the regularised cost function. Two new algorithms are introduced by considering two different weighting scenarios for the l(1) norm penalty. These new l(1) relaxation-based RLS algorithms emphasise sparsity during the adaptive filtering process, and they allow for faster convergence than standard RLS when the system under consideration is sparse. The authors test the performance of the novel algorithms and compare it with standard RLS and other adaptive algorithms for sparse system identification.