Finite-time contractivity profiling of a two-parameter parallel root-finding scheme via a kNN-LLE proxy

Abstract

Parallel iterative schemes are widely used for the simultaneous computation of all distinct roots of nonlinear equations in scientific computing and engineering. While highorder 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 problemdependent, expensive to scale, and difficult to automate reproducibly. In this work, we propose a data-driven framework for systematic parameter optimization based on finitetime 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.