  As a basic example in nonlinear theories of discrete complex analysis, we
explore various numerical methods for the accurate evaluation of the discrete
map $Z^a$ introduced by Agafonov and Bobenko. The methods are based either on a
discrete Painlev\'e equation or on the Riemann-Hilbert method. In the latter
case, the underlying structure of a triangular Riemann-Hilbert problem with a
non-triangular solution requires special care in the numerical approach.
Complexity and numerical stability are discussed, the results are illustrated
by numerical examples
