A practical strategy to improve performance of Newton's method in solving nonlinear equations

Abstract

Newton's method is a popular numeric approach due to its simplicity and quadratic convergence to solve nonlinear equations that cannot be solved with exact solutions. However, the initial point chosen to activate the iteration of Newton's method may cause difficulties in slower convergence, stagnation, and divergence of the iterative process. The common advice to deal with these special cases was to choose another inner point to repeat the process again or use a graph of the nonlinear equation (function) to choose a new initial point. Based on the recent experiences in teaching preservice secondary-school mathematics teachers, this classroom note presents a simple and practical strategy to avoid many of the difficulties encountered in using Newton's method during solving nonlinear equations. Instead of plotting the graph of an equation to help choose the initial point, the practical strategy is to use the middle point of the range as the initial point to start the iterative process of Newton's method. By solving ten different nonlinear equations using Newton's method initiated from the endpoints and the middle point respectively, the results show that choosing the middle point of a defined range to initiate the iterative process for Newton's method seems a general and practical strategy to avoid the difficulties encountered in using the method to solve nonlinear equations with faster convergence in most cases.