A comprehensive study of Newton's method, Halley's method, Super-Halley's method and Modified Super-Halley's method for solving nonlinear equations

This project analyses the efficiency of Newton's method, Halley's method, Super­Halley's method and Modified Super-Halley's method for solving nonlinear equations. The objective of this study is to identify the most effective method to solve nonlinear equations. These methods were tested using 10 nonlinear test functions consisting of combination of polynomial, trigonometric, logarithmic and exponential functions. The algorithm was tested under different initial guesses and measured using two tolerance levels. These nonlinear functions were executed using MAPLE 2016 software to determine their performance based on number of iterations and CPU time. SigmaPlot was then used to generate performance profiles to compare and analyse the graphs based on number of iterations and CPU time. This project observed that Super-Halley's method is the best performing method in terms of number of iterations. Super-Halley's method successfully solved all of the test functions with the fewest calculation steps. However, Newton's method is the fastest in terms of CPU time. Newton's method successfully solved all the test functions with the least computational cost. This project highlights that using higher-order method such as Modified Super-Halley's method does not guarantee the most efficient performance because higher-order method is highly sensitive to initial guesses and computationally expensive.