  For nonlinear equations, the homotopy methods (continuation methods) are
popular in engineering fields since their convergence regions are large and
they are quite reliable to find a solution. The disadvantage of the classical
homotopy methods is that their computational time is heavy since they need to
solve many auxiliary nonlinear systems during the intermediate continuation
processes. In order to overcome this shortcoming, we consider the special
explicit continuation Newton method with the residual trust-region
time-stepping scheme for this problem. According to our numerical experiments,
the new method is more robust and faster to find the required solution of the
real-world problem than the traditional optimization method (the built-in
subroutine fsolve.m of the MATLAB environment) and the homotopy continuation
methods(HOMPACK90 and NAClab). Furthermore, we analyze the global convergence
and the local superlinear convergence of the new method.
