Recently Van Assche (in J. Diff. Equ. Appl. 30, 465-474, 2024) showed the existence and uniqueness of a special solution of a discrete Painleve II equation, which arises in connection with certain orthogonal polynomials on the unit circle, and are expressed explicitly in terms of a ratio of modified Bessel functions. Here we consider a fixed point iteration on semi-infinite sequences of real numbers in the interval [−1, 1], which converges to this unique solution of discrete Painleve II, and provides an effective numerical scheme for computing these classical solutions.