Local stability theory for fixed-point iterations provides classical conditions for convergence, yet in scientific computing these conditions are rarely used to actively interpret or guide algorithmic behavior. Instead, convergence properties are commonly inferred from observed iteration traces, an approach that can be unreliable under finite-precision arithmetic. This work presents a computational study showing that local stability quantities remain predictive in practice, but only when interpreted in conjunction with the numerical regime governing the computation. Using infinite tetration as a minimal and analytically tractable fixed-point system, we systematically compare theoretical stability predictions with observed iteration behavior in floating-point arithmetic. Although classical stability theory correctly describes the latent dynamics of the iteration, we demonstrate that finite precision, rapid convergence, and degeneracies at the fixed point can suppress, distort, or falsely indicate instability in observed iterates. We identify three numerically distinct regimes—precision-dominated, transient-starved, and degenerate fixed-point regimes—in which naı̈ve interpretation of iteration histories leads to systematic misclassification of convergence behavior. Our results suggest that local stability measures, such as the derivative at the fixed point, should be treated as computable diagnostics rather than passive convergence guarantees. When combined with numerical regime awareness, these diagnostics reliably predict convergence mode and iteration difficulty, and provide principled guidance for stopping, damping, or modifying iterative algorithms. Although infinite tetration serves as an illustrative system, the conclusions apply broadly to fixed-point iterations and root-finding methods commonly used in scientific computing.
Taiwo Megbope (Thu,) studied this question.