Recently sparse representation has been applied to visual tracker by modeling the target appearance using a sparse approximation over a template set, which leads to the so-called L_1 trackers as it needs to solve an ℓ_1 norm related minimization problem for many times. While these L_1 trackers showed impressive tracking accuracies, they are very computationally demanding and the speed bottleneck is the solver to ℓ_1 norm minimizations. This paper aims at developing an L_1 tracker that not only runs in real time but also enjoys better robustness than other L_1 trackers. In our proposed L_1 tracker, a new ℓ_1 norm related minimization model is proposed to improve the tracking accuracy by adding an ℓ_1 norm regularization on the coefficients associated with the trivial templates. Moreover, based on the accelerated proximal gradient approach, a very fast numerical solver is developed to solve the resulting ℓ_1 norm related minimization problem with guaranteed quadratic convergence. The great running time efficiency and tracking accuracy of the proposed tracker is validated with a comprehensive evaluation involving eight challenging sequences and five alternative state-of-the-art trackers.