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Blind equalization is widely utilized to eliminate inter-symbol interference in communication system. It is still a challenge to support blind equalization method within non-Gaussian noise environments. Aiming at improving its convergence speed and robustness performance, in this letter, a novel fast blind equalization method is proposed by using bounded nonlinear function (BNF) and quasi-Newton method. Firstly, BNF-based cost function is proposed to effectively eliminate non-Gaussian noise and realize equalization. Next, quasi-Newton method is developed as the iteration method, which can accelerate the convergence speed. Moreover, theoretical analysis is provided to illustrate that the proposed algorithm has a robust convergence performance. Simulation results show that the proposed method enables evident performance improvement in terms of convergence speed and robustness with non-Gaussian noise.