New ideas to the design of algorithms based on derivatives
Flavio Barbosa, Fernando Nogueira
TL;DR
This work introduces a Generalized Derivative framework to extend derivative-based numerical methods, notably the Newton-Raphson and Gradient methods, with the aim of reducing iterations via higher-order interpolants encoded by Derivator Functions. By substituting Linear with Quadratic or Cubic Derivatives, the Quadratic G (Q-G) and Cubic NR (C-NR) methods achieve faster convergence and greater robustness, as illustrated through geometric interpretations and a polynomial example. The Q-G and C-NR algorithms demonstrate significantly fewer iterations and reduced sensitivity to initial guesses compared to classical counterparts, highlighting the practical potential of generalized differentiation for root finding and extrema optimization. The paper also outlines multiple avenues for future research, including multivariate generalizations, convergence theory, and the exploration of non-Euclidean Derivator Functions to broaden applicability and theoretical guarantees.
Abstract
This article proposes new perspectives for developing derivative based numerical algorithms, supported by the introduction of a generalized derivative operators. It demonstrates that these operators have the potential to enhance and extend existing derivativebased numerical methods. To this end, two iterative derivative driven methods are examined and refined: the Newton Raphson method and the Gradient method. For both approaches, generalized derivatives are introduced with the goal of reducing the number of iterations required for convergence. These modifications are presented through geometric interpretations of the proposed constructions, which clearly illustrate their convergenceaccelerating properties. The concluding remarks emphasize the significant opportunity to advance and refine numerical algorithms through the use of generalized derivatives.