Penentuan Semua Akar Real Polinomial Secara Serentak Menggunakan Integrasi Metode Evolusi Diferensial-Cauchy-Niching Dan Newton Polishing

Abstract

Determining all real roots of high-degree polynomials remains a challenge in numerical analysis, particularly when roots are adjacent or clustered. Conventional methods, such as Newton-Raphson, require precise initial guesses and tend to fail when roots are adjacent, often converging to only one dominant root. This study proposes an integration of three complementary methods for the simultaneous determination of all real roots of polynomials: (a) Cauchy Bound to automatically determine the search space, (b) Differential Evolution (DE) with a Niching mechanism for global exploration and simultaneous localization of all roots, and (c) Newton Polishing to enhance root accuracy. The Differential Evolution parameters employed are mutation factor F = 0.5 and crossover probability Cr = 0.9. The method was tested on three fifth-degree polynomials with distinct characteristics: large coefficients (P1), decimal coefficients (P2), and adjacent roots separated by 0.1 (P3). Simulation results demonstrate that the proposed method successfully identified all real roots of the three test polynomials in a single execution, without requiring multiple initial guesses. The average computation time was less than 5 seconds per polynomial. This integrated approach proves more effective than the conventional Newton-Raphson method, which requires initial guesses and remains prone to failure when roots are adjacent.