The general Newton method proposed in \cite{NF} is $q$-linear convergent. We define a modification of the general Newton method. The main idea of this method is to use a given number of inner iterations for calculating approximation of the inverse Jacobian matrix, because it has an important influence on the convergence rate of the method. Local $q$-linear, $q$-superlinear and $q$-quadratic convergence of the modification of the general Newton method is proved. Some numerical experiments are also presented.