Abstract
We study early prediction of convergence outcomes in parameterized root-finding problems. The analysis focuses on whether short prefixes of solver trajectories contain useful information about convergence within prescribed iteration horizons, without modifying the underlying numerical scheme. To avoid relying on profile-derived targets, we use solver-level success fractions (Formula presented.), (Formula presented.), and (Formula presented.), defined as the fractions of trajectories that satisfy the convergence criterion within 20, 50, and 100 iterations, respectively. Within this setting, we compare several families of early solver-derived features, including residual-based, step-based, and trajectory-derived quantities. We also distinguish between direct-prefix predictors, which use the first N values of a feature family, and scalar summary predictors, in which the same prefix is compressed into a single descriptor. Numerical experiments on controlled parameterized root-finding benchmarks show that simple residual- and step-based summaries provide the strongest within-base predictors among the tested feature families. In particular, mean and median log residual and log step summaries are informative even at short prefixes. More elaborate trajectory-derived descriptors, including the kNN–LLE proxy, are less effective as standalone predictors in the present experiments. The results also show that prediction difficulty depends strongly on the benchmark structure: Dataset 1 leads to near-saturated prediction performance at very short prefixes, whereas the oscillatory Dataset 2A provides a more challenging and discriminative case. In this latter setting, compact scalar summaries at (Formula presented.) can occasionally match or slightly outperform the corresponding full-prefix predictors, suggesting that scalar compression may suppress part of the transient variability present in the full-prefix representation. Cross-base transfer is substantially more difficult than within-base prediction, indicating that the observed predictive relationships are not uniformly portable across the considered benchmark problems. Overall, the study suggests that, in controlled parameterized settings of the type examined here, convergence outcomes can often be predicted from simple and computationally inexpensive early trajectory summaries. The results should be interpreted as a feature-based early convergence outcome prediction study, rather than as a general diagnostic methodology for nonlinear solvers.