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

This paper considers the generalized continuation Newton method and the trust-region updating strategy for the underdetermined system of nonlinear equations.Moreover, in order to improve its computational efficiency, the new method uses a switching updating technique of the Jacobian matrix. That is to say, it does not compute the next Jacobian matrix and replaces it with the current jacobian matrix when the linear approximation model of the merit function approximates it 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 environment). The computational speed of the new method is about eight to fifty times as fast as that of fsolve. Furthermore, it also proves the global convergence and the local superlinear convergence of the new method under some standard assumptions.