Recently, several mathematical results and techniques related to the so-called Nevanlinna-Pick interpolation problem have found important applications in a variety of Control/System theoretic problems. The solution of the Nevanlinna-Pick problem is obtained in an inductive way through the so-called Schur algorithm. In earlier work, we have exploited the Schur algorithm as a means to achieve spectral factorization. The present paper is a continuation of that work, in particular, we investigate the Schur algorithm in the context of the tangential Nevanlinna-Pick problem. This provides a computationally simple scheme for spectral factorization of matrix-valued functions.