Long-term optimality of AAA versus alternative greedy rational approximation methods

Ascertain whether the AAA algorithm will remain the best approach for greedy rational approximation across applications when compared to alternatives such as Thiele continued fractions, by establishing criteria and empirical or theoretical evidence for long-term superiority or equivalence.

Background

The paper reviews the AAA algorithm alongside other greedy rational approximation methods, notably Thiele continued fractions (TCF), highlighting recent evidence that TCF can achieve similar speed and accuracy in many cases. While AAA has enabled widespread practical success, the authors explicitly state uncertainty about its long-term dominance. This problem invites comparative analyses and theoretical frameworks to evaluate and predict method performance across diverse application classes.