Numerical Methods for the Discrete Map $Z^a$

TL;DR

This paper investigates numerical techniques for accurately computing the discrete map $Z^a$, comparing approaches based on discrete Painlevé equations and Riemann-Hilbert problems, addressing stability and complexity issues.

Contribution

It introduces and analyzes numerical methods for $Z^a$, focusing on handling complex structures like non-triangular Riemann-Hilbert problems.

Findings

Methods based on Painlevé equations are effective for certain parameter ranges.

Riemann-Hilbert approach requires careful numerical treatment due to structural complexities.

Numerical stability and complexity are thoroughly discussed.

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 $Z^a$ 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