An efficient recursive solution is presented for the one-sided unconstrained tangential interpolation problem, The method relies on the triangular factorization of a certain structured matrix that is implicitly defined by the interpolation data. The recursive procedure admits a physical interpretation in terms of discretized transmission lines, In this framework the generating system is constructed as a cascade of first-order sections, Singular steps occur only when the input data is contradictory, i.e., only when the interpolation problem does not have a solution, Various pivoting schemes can be used to improve numerical accuracy or to impose additional constraints on the interpolants, The algorithm also provides coprime factorizations for all rational interpolants and can be used to solve polynomial interpolation problems such as the general Hermite matrix interpolation problem. A recursive method is proposed to compute a column-reduced generating system that can be used to solve the minimal tangential interpolation problem.