Analisis Perbandingan Metode Biseksi dan Metode Newton-Raphson dalam Menentukan Akar Persamaan Non Linear

Abstract

This study aims to compare two numerical methods for finding roots of nonlinear equations, namely the Bisection Method and the Newton-Raphson Method, with a case study on the function f(x) = x³ − 2x − 5 = 0. This is a comparative quantitative study focusing on four main parameters: number of iterations, error rate, root accuracy, and convergence speed. The Bisection Method is implemented with an initial interval [2, 3] based on the Intermediate Value Theorem, while the Newton-Raphson Method uses an initial point x₀ = 2.5 utilizing the derivative f'(x) = 3x² − 2. Results show that the Newton-Raphson Method achieves convergence in 4 iterations with a root value of x ≈ 2.094551 and a final error less than 10⁻⁶, while the Bisection Method requires 13 iterations with a root value of x ≈ 2.094604 at tolerance ε = 10⁻⁴. The Newton-Raphson Method exhibits quadratic convergence (p = 2), which is significantly faster than the linear convergence of the Bisection Method (p = 1). Nevertheless, the Bisection Method maintains advantages in stability and simplicity as it does not require function derivatives. The conclusion is that the Newton-Raphson Method is more computationally efficient for polynomial functions with easily computable derivatives, while the Bisection Method is recommended when functions are difficult to differentiate or when stability is the priority.