The authors present a systematic, recursive solution of tangential matrix interpolation problems of Hermite-Fejer, Nevanlinna-Pick type. The core of the approach is an algorithm for the recursive triangular factorization of matrices with displacement structure which the authors and various colleagues have been developing for a number of years in the context of other applications. Use of this algorithm leads to a recursive construction of the so-called resolvent matrix for the interpolation problem which provides a linear fractional parametrization for the set of all solutions. The calculations are so arranged that one computes the resolvent matrix for the problem with one interpolation node added by a simple one-step update of the solution already at hand rather than having to start over again. The algorithm can be interpreted as a recursive updating of a Cholesky factorization of a nested family of Pick matrices associated with the interpolation data, and as exhibiting a factorization of the resolvent matrix for the whole problem (a \(J\)-inner matrix determined by the interpolation data) into elementary Blaschke-Potapov factors.