Finite-Time Contractivity Profiling of a Two-Parameter Parallel Root-Finding Scheme via a kNN–LLE Proxy

Parallel iterative schemes are widely used for the simultaneous computation of all distinct roots of nonlinear equations in scientific computing and engineering. While high-order parallel methods can provide substantial acceleration, their practical performance is often dominated by the choice of internal real-valued parameters introduced by correction/acceleration mechanisms, which may strongly affect convergence speed and numerical robustness. Classical parameter-selection strategies—based on analytical sufficient conditions, trial-and-error experimentation, or qualitative dynamical diagnostics (basins of attraction, bifurcation-style inspection, and parameter planes)—are typically problem-dependent, expensive to scale, and difficult to automate reproducibly. In this work, we propose a data-driven framework for systematic parameter optimization based on finite-time contractivity profiling. The approach uses k-nearest neighbors (kNN) micro-series analysis to estimate a proxy profile of the largest Lyapunov exponent (LLE) along the iteration index, summarizing the transient contraction/expansion behavior of the solver trajectories. Two profile-based scores, the minimum score Smin and the moment score Smom, are introduced to rank candidate parameter pairs and to construct stability landscapes over (α,β) grids. As a testbed, we apply the framework to a bi-parametric two-step parallel Weierstrass-type scheme and demonstrate that the learned parameter regions yield faster and more reliable convergence than generic or manually tuned choices. Extensive numerical experiments show that the proposed profiling-based optimization consistently improves convergence rate and robustness across the considered nonlinear test problems, providing a scalable and reproducible alternative to heuristic and dynamical-system-based tuning.