Efficient continuation Newton-like method for solving systems of non-linear equations

In this paper, we present a new Newton-like method for finding a zero of a vector function, permitting that the Jacobian is singular in some points. Thus, the problems due to the fact the Jacobian is numerically singular are solved. The method is showed to be quadratic convergence. Then, we detail the continuation Newton-like method corresponding to this method. By numerical examples, we show that this method is vast superior and possible in global convergence.