Evaluación del Rendimiento del Método Resolvente Encadenado (MRLBS): Un Algoritmo Estructuralmente Consciente para la Resolución Simbólica de Polinomios
Performance Evaluation of the Chain Resolution Method (MRLBS): A Structurally Aware Algorithm for Symbolic Polynomial ResolutionContenido principal del artículo
La eficiencia en la resolución simbólica de polinomios de grado superior es un desafío persistente en el álgebra computacional, donde los métodos de propósito general a menudo no explotan la estructura interna del problema. Este artículo introduce y evalúa empíricamente el Método Resolvente Encadenado de Lagrange-Bring-Sánchez (MRLBS), un nuevo algoritmo fundamentado en la Teoría Algebraica de Resolventes Encadenadas (TGRAE) que aprovecha la factorización estructural guiada por la teoría de Galois. Se realizó un estudio de rendimiento comparativo frente a dos solvers de referencia utilizando un corpus estratificado de 150 polinomios. El análisis estadístico, mediante un ANOVA de dos factores, reveló un efecto de interacción significativo (F (4, 441) = 8.92, p < .001, ηp² = 0.075), demostrando que la ventaja de rendimiento de MRLBS no solo es significativa, sino que se magnifica a medida que aumenta el grado del polinomio. Estos hallazgos, robustos tras controlar por la densidad de términos, validan cuantitativamente la eficacia de un enfoque estructuralmente consciente. Este trabajo no solo presenta un algoritmo superior y escalable, sino que aboga por una integración más profunda de la teoría algebraica abstracta en el diseño de herramientas computacionales de alto rendimiento.
Efficiency in the symbolic resolution of higher-degree polynomials remains a persistent challenge in computational algebra, where general-purpose methods often fail to exploit the problem's internal structure. This paper introduces and empirically evaluates the Lagrange-Bring-Sánchez Chained Resolvent Method (MRLBS), a novel algorithm based on the Algebraic Theory of Chained Resolvents (TGRAE) that leverages structural factorization guided by Galois theory. A comparative performance study was conducted against two benchmark solvers using a stratified corpus of 150 polynomials. Statistical analysis, through a two-way ANOVA, revealed a significant interaction effect (F(4, 441) = 8.92, p < .001, ηp² = 0.075), demonstrating that MRLBS's performance advantage is not only significant but also magnifies as the polynomial degree increases. These findings, robust after controlling for term density, quantitatively validate the effectiveness of a structurally-aware approach. This work not only presents a superior and scalable algorithm but also advocates for a deeper integration of abstract algebraic theory into the design of high-performance computational tools.
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