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This work is concerned with the computation of $\ell^p$-eigenvalues andeigenvectors of square tensors with $d$ modes. In the first part we propose twopossible shifted variants of the popular (higher-order) power method %for thecomputation of $\ell^p$-eigenpairs proving the convergence of both the schemesto the Perron $\ell^p$-eigenvector of the tensor, and the maximal corresponding$\ell^p$-eigenvalue, when the tensor is entrywise nonnegative and $p$ isstrictly larger than the number of modes. Then, motivated by the slow rate ofconvergence that the proposed methods achieve for certain real-world tensors,when $p\approx d$, the number of modes, in the second part we introduce anextrapolation 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 improvementsgained by the introduction of the shifting parameter and the efficiency of theacceleration technique.

Shifted and extrapolated power methods for tensor $\ell^p$-eigenpairs