  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.
