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In this paper, we investigate the performance of positioning algorithms based on time difference of arrival (TDOA) measurements provided by the base stations or ENodeBs in LTE (Long Term Evolution) context. Using the TDOA approach the localization process of the mobile User equipment results in a non-linear least squares estimation problem which cannot be solved analytically. Therefore, there is different algorithms that could be used to determine an estimate of the mobile station position. As for the Gauss-Newton method, it fails to converge for certain geometric constellations, and thus, it is not suitable for a general solution in cellular networks. Another algorithm is the steepest descent method which has a slow convergence in the final iteration steps. Hence, we apply the Levenberg-Marquardt algorithm as a new approach in the LTE network localization framework. We show that this method meets the best trade-off between accuracy and computational complexity by a comparative study deploying the three algorithms in the same simulation environment and investigating their performances.