In this work, we propose an alternating low-rank decomposition (ALRD) approach and novel subspace algorithms for direction-of-arrival (DOA) estimation. In the ALRD scheme, the decomposition matrix for rank reduction is composed of a set of basis vectors. A low-rank auxiliary parameter vector is then employed to compute the output power spectrum. Alternating optimization strategies based on recursive least squares (RLS), denoted as ALRD-RLS and modified ALRD-RLS (MARLD-RLS), are devised to compute the basis vectors and the auxiliary parameter vector. Simulations for large sensor arrays with both uncorrelated and correlated sources are presented, showing that the proposed algorithms are superior to existing techniques.