Various approaches to solving nonlinear equations

TL;DR

This work surveys and benchmarks multiple solver paradigms for nonlinear equations, nonlinear least-squares, and general function minimization in R, using the Dennis–Gay–Vu test problems to illustrate practical performance. It compares direct solvers (nleqslv, dfsane variants), stabilized LS methods (nlsr), and global/stochastic strategies (optimx, gloptim), highlighting when each approach succeeds or struggles, particularly in the presence of near-singular Jacobians. Key findings show that nleqslv often provides the fastest route to zero residuals, while function-minimization and stochastic methods frequently fail to reach exact solutions or require substantial tuning. The study emphasizes the importance of problem formulation (full vs. reduced DGV) and diagnostics such as Jacobian singular values and residual scaling for reliable solution of nonlinear systems in practice.

Abstract

Modelling real world systems frequently requires the solution of systems of nonlinear equations. A number of approaches have been suggested and developed for this computational problem. However, it is also possible to attempt solutions using more general nonlinear least squares or function minimization techniques. There are concerns, nonetheless, that we may fail to find solutions, or that the process will be inefficient. Examples are presented with R with the goal of providing guidance on the solution of nonlinear equations problems.