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In this paper, we propose an efficient algorithm to accelerate the existing Broad Learning System (BLS) algorithm for new added nodes. The existing BLS algorithm computes the output weights from the pseudoinverse with the ridge regression approximation, and updates the pseudoinverse iteratively. As a comparison, the proposed BLS algorithm computes the output weights from the inverse Cholesky factor of the Hermitian matrix in the calculation of the pseudoinverse, and updates the inverse Cholesky factor efficiently. Since the Hermitian matrix in the definition of the pseudoinverse is smaller than the pseudoinverse, the proposed BLS algorithm can reduce the computational complexity, and usually requires less than $\frac {2}{3}$ of complexities with respect to the existing BLS algorithm. Our experiments on the Modified National Institute of Standards and Technology (MNIST) dataset show that the speedups in accumulative training time and each additional training time of the proposed BLS over the existing BLS are 24.81%~ 37.99% and 36.45%~ 58.96%, respectively, and the speedup in total training time is 37.99%. In our experiments, the proposed BLS and the existing BLS both achieve the same testing accuracy when the tiny differences (≤ 0.05%) caused by the numerical errors are neglected, and the above-mentioned tiny differences and numerical errors become zeroes and ignorable, respectively, when the ridge parameter is not too small.