Survey of Methods for Solving Systems of Nonlinear Equations, Part I: Root-finding Approaches

Abstract

This paper presents a comprehensive survey of methods which can be utilized to search for solutions to systems of nonlinear equations (SNEs). Our objectives with this survey are to synthesize pertinent literature in this field by presenting a thorough description and analysis of the known methods capable of finding one or many solutions to SNEs, and to assist interested readers seeking to identify solution techniques which are well suited for solving the various classes of SNEs which one may encounter in real world applications.To accomplish these objectives, we present a multi-part survey. In part one, we focus on root-finding approaches which can be used to search for solutions to a SNE without transforming it into an optimization problem. In part two, we will introduce the various transformations which have been utilized to transform a SNE into an optimization problem, and we discuss optimization algorithms which can then be used to search for solutions. In part three, we will present a robust quantitative comparative analysis of methods capable of searching for solutions to SNEs.

Parallel GPU-Acceleration of Metaphorless Optimization Algorithms: Application for Solving Large-Scale Nonlinear Equation Systems

“…The well-known Newton method and its variants [6] are among the most common iterative numerical techniques used to solve NESs. The effectiveness of these solvers is strongly dependent on the quality of the initial approximations used [7,8], and convergence may not be guaranteed.…”