  This work is concerned with the computation of $\ell^p$-eigenvalues and
eigenvectors of square tensors with $d$ modes. In the first part we propose two
possible shifted variants of the popular (higher-order) power method %for the
computation of $\ell^p$-eigenpairs proving the convergence of both the schemes
to the Perron $\ell^p$-eigenvector of the tensor, and the maximal corresponding
$\ell^p$-eigenvalue, when the tensor is entrywise nonnegative and $p$ is
strictly larger than the number of modes. Then, motivated by the slow rate of
convergence that the proposed methods achieve for certain real-world tensors,
when $p\approx d$, the number of modes, in the second part we introduce an
extrapolation framework based on the simplified topological
$\varepsilon$-algorithm to efficiently accelerate the shifted power sequences.
Numerical results on synthetic and real world problems show the improvements
gained by the introduction of the shifting parameter and the efficiency of the
acceleration technique.
