Stepsize control for Newton's method in the presence of singularities

TL;DR

This paper explores stepsize control techniques for Newton's method to effectively detect and handle singularities in the Jacobian matrix, improving convergence near problematic points.

Contribution

It demonstrates that existing stepsize control strategies can be adapted to identify and rapidly converge to Jacobian singularities in Newton's method.

Findings

Stepsize controls can detect Jacobian singularities.

Rapid convergence to singularities is achievable.

Enhances robustness of Newton's method near singular points.

Abstract

Singularities in the Jacobian matrix are an obstacle to Newton's method. We show that stepsize controls suggested by Deuflhard and Steinhoff can be used to detect and rapidly converge to such singularities.