Exploring equations ordering influence on variants of the Newton-Raphson method

Jacobian-free Newton-Raphson methods are general purpose iterative non-linear system solvers. The need to solve non-linear systems is ubiquitous throughout computational physics [1] and Jacobian-free Newton-Raphson methods can offer scalability, super-linear convergence and applicability. In fact, applications span from discretized PDEs [2] to power-flow problems [3]. The focus of this article is on Inexact-Newton-Krylov [2] and Quasi-Inverse-Newton [4] methods. For both of them, we prove analytically that the initial ordering of the equations can have a great impact on the numerical solution, as well as on the number of iterations to reach the solution. We also present numerical results obtained from a simple but representative case study, to quantify the impact of initial equations ordering on a concrete scenario.