Newton’s Methods: Constraints and Higher Order Convergence

Abstract

Newton’s method is designed to approximate the roots of functions with iterative process. The basic idea of Newton’s method is repeatedly calculating the next approximation with a favorable initial guess which has a tangent line. Under some cases, such as indifferentiable functions, Newton’s method does not work. Since Newton’s method is not always desirable, it is varied to fit in different situations, and hence, different variants of Newton’s method prosper. One kind of variants aims to enhance the speed and the efficiency of Newton’s method, which is mainly discussed in this paper. Also, this kind of variants extends the applicability in different practical problems like chemicals and increase the order of convergence. This paper also discusses the efficiency of variants with efficiency index to normalize different methods and finds out real world problems need solutions with higher-order convergence and less computational cost. Nowadays, Newton’s method and its variants are still a powerful tool in diverse scientific applications.