Generalized continuation Newton methods and the trust-region updating strategy for the underdetermined system

TL;DR

This paper introduces a generalized continuation Newton method with a trust-region strategy for underdetermined nonlinear systems, improving computational efficiency and robustness over traditional methods like Levenberg-Marquardt.

Contribution

It proposes a new method that avoids Jacobian updates when unnecessary, enhancing speed and robustness, with proven global and local convergence properties.

Findings

The new method is significantly faster, about 1/8 to 1/50 of fsolve's time.

It demonstrates higher robustness in solving underdetermined systems.

The method has proven global convergence and local superlinear convergence.

Abstract

This paper considers the generalized continuation Newton method and thetrust-region updating strategy for the underdetermined system of nonlinear equations. Moreover, in order to improve its computational efficiency, the new method will not update the Jacobian matrix when the current Jacobian matrix performs well. The numerical results show that the new method is more robust and faster than the traditional optimization method such as the Levenberg-Marquardt method (a variant of trust-region methods, the built-in subroutine fsolve.m of the MATLAB R2020a environment). The computational time of the new method is about 1/8 to 1/50 of that of fsolve. Furthermore, it also proves the global convergence and the local superlinear convergence of the new method under some standard assumptions.