Application of polynomial algebras to non-linear equation solvers
For researchers in numerical analysis and computational science, this offers a novel way to enhance classical solvers with higher-order accuracy, though the impact is domain-specific.
The paper introduces Jet Transport, a high-order automatic differentiation technique, to Newton's method, proving it doubles correct Taylor coefficients per iteration and achieves quadratic convergence. The Jet-Newton method is validated on Kepler's equation and a three-body problem, providing high-order semi-analytical approximations.
This paper presents a novel application of Jet Transport, a high-order automatic differentiation technique, to enhance classical numerical methods, with a focus on Newton's method. We prove a central theorem establishing that, under appropriate conditions, applying Jet Transport within a Newton iteration doubles the number of correct coefficients in the Taylor series approximation of the solution. This theoretical result is then extended to the practical case where the exact solution is unknown, demonstrating the expected quadratic convergence (error reduction from \( \varepsilon \) to \( \varepsilon^2 \)) while simultaneously doubling the order of accuracy in the series expansion. The efficacy of the resulting Jet-Newton method is demonstrated through three illustrative examples: an academic problem validating the theoretical convergence rates, the solution of Kepler's equation, and a new continuation algorithm for computing zero-velocity curves in the circular restricted three-body problem. These examples showcase the method's capability to provide high-order semi-analytical approximations.