Root finding techniques that work

TL;DR

This paper introduces adaptive root-finding techniques that leverage the specific structure of nonlinear equations, improving efficiency over traditional all-purpose methods through problem-specific tailoring and transformations.

Contribution

It presents general techniques for customizing and transforming nonlinear equations to enhance root-finding efficiency, filling a gap in standard numerical analysis education.

Findings

Techniques for problem-specific method construction

Transformations to simplify equations for existing methods

Case studies demonstrating improved convergence

Abstract

In most introductory numerical analysis textbooks, the treatment of a single nonlinear equation often consists of a collection of all-purpose methods that frequently do not work or are inefficient. These textbooks neglect to teach the importance of adapting a method to the given problem, and consequently also neglect to provide the tools to accomplish this. Several general techniques are described here to incorporate the specific structure or properties of a nonlinear equation into a method for solving it. This can mean the construction of a method specifically tailored to the equation, or the transformation of the equation into an equivalent one for which an existing method is well-suited. The techniques are illustrated with the help of several case studies taken from the literature.