  We prove a sharp stability estimate for Schur iterates of contractive
analytic functions in the open unit disk. We then apply this result in the
setting of the inverse scattering approach and obtain a fast algorithm for
solving the discrete integrable nonlinear Schr\"odinger equation
(Ablowitz-Ladik equation) on the integer lattice, $\mathbb{Z}$. We also give a
self-contained introduction to the theory of the nonlinear Fourier transform
from the perspective of Schur functions and orthogonal polynomials on the unit
circle.
