The multilinear PageRank is an extension of the well-known PageRank model. The solution of this model comes as a Z-eigenvector of a non-negative tensor. High-order power method is one of the most widely used ways of computing the multilinear PageRank vector. Even for irreducible and aperiodic tensors, the approach may not converge and when it converges, the convergence may be slow. For larger problems, these two limitations make computing the eigenvectors difficult or impossible. The paper proposes a new method for accelerating the computation of the multilinear PageRank vector using a vector Aitken extrapolation method.