A third-order modification of Newton method for systems of non-linear equations

In this paper, we present a new modified Newton method for finding a zero of a vector function. It is proved that the order of convergence of the new method is three. This new method permits that the Jacobian is singular in some points. Thus the expected problem, due to the fact the Jacobian is numerically singular, is solved. By numerical examples, we show that this method is vast superior in some cases.