Summary Nonnegative tensors arise very naturally in many applications that involve large and complex data flows. Due to the relatively small requirement in terms of memory storage and number of operations per step, the (shifted) higher order power method is one of the most commonly used technique for the computation of positive Z‐eigenvectors of this type of tensors. However, unlike the matrix case, the method may fail to converge even for irreducible tensors. Moreover, when it converges, its convergence rate can be very slow. These two drawbacks often make the computation of the eigenvectors demanding or unfeasible for large problems. In this work, we consider a particular class of nonnegative tensors associated with the multilinear PageRank modification of higher order Markov chains. Based on the simplified topological ε‐algorithm in its restarted form, we introduce an extrapolation‐based acceleration of power method type algorithms, namely, the shifted fixed‐point method and the inner‐outer method. The accelerated methods show remarkably better performance, with faster convergence rates and reduced overall computational time. Extensive numerical experiments on synthetic and real‐world datasets demonstrate the advantages of the introduced extrapolation techniques.

This work is concerned with the computation of ℓ 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 and, when the tensor is entrywise nonnegative with a possibly reducible pattern and p is strictly larger than the number of modes, we prove the convergence of both the schemes to the Perron ℓ p -eigenvector and to the maximal corresponding ℓ p -eigenvalue of the tensor. Then, in the second part, motivated by the slow rate of convergence that the proposed methods achieve for certain realworld tensors when p ≈ d, the number of modes, we introduce an extrapolation framework based on the simplified topological ε-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.

In this paper we study adaptive L (k) QN methods, involving special matrix algebras of low complexity, to solve general (non-structured) unconstrained minimization problems. These methods, which generalize the classical BFGS method, are based on an iterative formula which exploits, at each step, an ad hoc chosen matrix algebra L (k) . A global convergence result is obtained under suitable assumptions on f .

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