An Efficient GIPM Algorithm for Computing the Smallest V-Singular Value of the Partially Symmetric Tensor

Real partially symmetric tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we try to compute the smallest V-singular value of partially symmetric tensors with orders (p, q). This is a unified notion in a broad sense that, when \((p,q)=(2,2)\) $(p,q)=(2,2)$ , the V-singular value coincides with the notion of M-eigenvalue. To do that, we propose a generalized inverse power method with a shift variable to compute the smallest V-singular value and eigenvectors. Global convergence of the algorithm is established. Furthermore, it is proven that the proposed algorithm always converges to the smallest V-singular value and the associated eigenvectors. Several numerical experiments show the efficiency of the proposed algorithm.