ANALISA PERFORMA METODE NEWTON-RAPHSON DAN ITERASI TITIK TETAP UNTUK MENYELESAIKAN AKAR SISTEM PERSAMAAN NON-LINIER
Abstract
Non-linear equations are one of the fundamental problems that often arise in various scientific disciplines, such as physics, engineering, economics and computer science. Solving non-linear equations analytically is often not possible due to the complex nature of the functions involved. Therefore, numerical methods such as Newton-Raphson and Fixed Point Iteration are the main choice to approach solutions with high accuracy. This research aims to analyze the performance of the two methods in solving systems of non-linear equations. The analysis is carried out by comparing aspects of convergence speed, solution accuracy, and stability to changes in initial values and the nature of the function being analyzed. The Newton-Raphson method is known for its fast quadratic convergence, but requires derivatives of functions that are not always practical to calculate. Meanwhile, the Fixed Point Iteration method is simpler in implementation, but has slower linear convergence and relies heavily on the selection of recursive functions and initial values. The research results show that the Newton-Raphson method is superior in terms of convergence speed, especially for functions with derivatives that can be calculated easily. On the other hand, Fixed Point Iteration is more flexible for use on functions without explicit derivatives, although it requires more iterations to achieve the same accuracy. This research provides guidance for practitioners in choosing the most appropriate numerical method based on the characteristics of the problems faced, so that it can be implemented optimally in various application fields.
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