We develop a spectral factorization algorithm based on linear fractional transformations and on the Nevanlinna-Pick interpolation theory. The algorithm is recursive and depends on a choice of points (z//k, k equals 1,2,. . . ) inside the unit disk. Under a mild condition on the distribution of the z//k's, the convergence of the algorithm is established. The algorithm is flexible and convergence can be influenced by the selection of z//k's.