Before revision:   The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is  not a saddle point, and the convergence rate is the same as that of Newton's method  .
After revision:  Here \delta _n is an appropriate real number so that A_n is invertible, and pr_{A_n,\pm The main result of this paper roughly says that if f is C^3 and a sequence \{x_n\}, constructed by the New Q-Newton's method from a random initial point x_0, {\bf converges}, then the limit point is not a saddle point, and the convergence rate is the same as that of Newton's method .