Numerical methods for the discrete map Za

Abstract

As a basic example in nonlinear theories of discrete complex analysis, we explore various numerical methods for the accurate evaluation of the discrete map Za introduced by Agafonov and Bobenko. The methods are based either on a discrete Painlevé 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.