On the choice of initial guesses for the Newton-Raphson algorithm

TL;DR

This paper investigates how the choice of initial guesses affects the convergence of the Newton-Raphson method for solving nonlinear equations, introducing criteria based on derivatives to improve initial guess selection.

Contribution

It introduces new derivative-based criteria to evaluate and improve initial guesses for Newton-Raphson, enhancing convergence reliability.

Findings

Criteria effectively identify problematic initial guesses.

Indicators based on derivatives predict convergence issues.

Test cases demonstrate practical applicability.

Abstract

The initialization of equation-based differential-algebraic system models, and more in general the solution of many engineering and scientific problems, require the solution of systems of nonlinear equations. Newton-Raphson's method is widely used for this purpose; it is very efficient in the computation of the solution if the initial guess is close enough to it, but it can fail otherwise. In this paper, several criteria are introduced to analyze the influence of the initial guess on the evolution of Newton-Raphson's algorithm and to identify which initial guesses need to be improved in case of convergence failure. In particular, indicators based on first and second derivatives of the residual function are introduced, whose values allow to assess how much the initial guess of each variable can be responsible for the convergence failure. The use of such criteria, which are based on rigorously proven results, is successfully demonstrated in three exemplary test cases.