Ebenezer's Algorithm: A Method for Solving Extreme Degree Polynomials in Seconds

This document presents Ebenezer’s algorithm, a modular, massively parallel GPU‑based root‑finding method designed to handle dense and ill‑conditioned polynomials with degrees up to \( n = 10^6 \) (and even multi‑GPU tests for \( 2 \times 10^6 \)). It combines: - GPU‑accelerated Aberth–Ehrlich iteration for simultaneous root approximation. - Geometric clustering of approximate roots via Schur trees built from Gershgorin disks. - Local Newton–Krylov refinement, explicit or Jacobian‑free, with adaptive preconditioning for stability. The paper details theoretical convergence (third‑order locally under standard assumptions), global complexity analysis, and extensive benchmarks against reference solvers like Jenkins–Traub and MPSolve — showing dramatic performance gains at very high degrees. Experimental scope: Benchmarks cover multiple polynomial families — Chebyshev, Legendre, Laguerre, Hermite, random monic, Wilkinson, and challenging quasi‑multiple‑root cases — with degrees \(10^3\) to \(10^6\). Metrics include: - Execution time per phase (GPU iteration, clustering, refinement) - Precision (max residual errors near machine epsilon for well‑conditioned cases) - GPU memory consumption and scalability trends - Impact of root distribution and conditioning on performance Key findings: - Near‑linear runtime scaling in practice due to high GPU parallel efficiency. - Robustness across difficult cases using adaptive clustering, under‑relaxation, and preconditioning. - Multi‑GPU and distributed extensions are viable for extreme problem sizes. - Applications include signal processing, cryptography, and quantum spectral modeling. Conclusion: The approach significantly outperforms classical sequential solvers at scale, maintains high accuracy, and adapts to a wide range of polynomial structures and cond itioning profiles. ---