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Alternating least squares (ALS) and its variations are the most commonly used algorithms for the PARAFAC decomposition of a tensor. However, it is still troubled for one how to accelerate the ALS algorithm with the reduced computational complexity. In this paper, a new acceleration method for the ALS with a matrix polynomial predictive model (MPPM) is proposed. In the MPPM, a matrix-valued function is first approximated by a matrix polynomial. It is shown that the future value of the function can be predicted by an FIR filter with the coefficients determined offline. By viewing each factor matrix of a tensor as a matrix-valued function, a new ALS algorithm, the ALS-MPPM algorithm, is then given. Analyses show that our ALS-MPPM algorithm is of low computational complexity and a close relation with the existing ALS algorithms. Moreover, to further accelerate the convergence of the proposed algorithm, a new technique called the multi-model (MM) prediction is also introduced. While the analytical results are verified by the numerical simulations, it is also shown that our ALS-MPPM outperforms the existing ALS-based algorithms in terms of the rate of convergence.