High-Performance Damped Traub-Type Iterative Scheme for Nonlinear Problems

We present a new family of high-performance schemes for solving non-linear systems. The method is simple and offers a very low computational cost. The convergence analysis shows that the order reaches twice the number of steps. We focus on a new eighth-order method that requires to solve only four linear systems per iteration. These systems share the same Jacobian matrix and a simple scalar weight function. Efficiency studies show that our method outperforms other optimal eighth-order and highly efficient ninth-order schemes in the literature. A dynamical analysis confirms its superior stability, showing large and connected convergence regions. Monte Carlo tests on the Bratu problem prove that the new scheme can exceed the stability of the optimal second-order Newton's method, introducing a new robustness test by using uncertainty vs. non-linearity. High-precision numerical tests on an elastic string problem validate the theoretical results. This work enables the practical use of very high-order schemes with minimal computational effort.