A practical method for computing the largest M‐eigenvalue of a fourth‐order partially symmetric tensor

Abstract

In this paper, we consider a bi-quadratic homogeneous polynomial optimization problem over two unit spheres arising in nonlinear elastic material analysis and in entanglement studies in quantum physics. The problem is equivalent to computing the largest M-eigenvalue of a fourth-order tensor. To solve the problem, we propose a practical method whose validity is guaranteed theoretically. To make the sequence generated by the method converge to a good solution of the problem, we also develop an initialization scheme. The given numerical experiments show the effectiveness of the proposed method.

The Best Rank-One Approximation Ratio of a Tensor Space

“…In the previous version of this paper, we got a positive lower bound [5], [11], [16], [20], [22], [25], [27].…”

“…We may also see that ρ B ðAÞ is the largest absolute value of the M -eigenvalues of A, defined as below [20], [25]. Denote A· xyy as a vector in ℜ n , whose ith component is P n j¼1 P p k;l¼1 a ijkl x j y k y l , and denote Axxy · as a vector in ℜ p , whose lth component is P n i;j¼1 P p k¼1 a ijkl x i x j y k .…”

“…We use the greedy update algorithm to decompose the tensors. In every iteration of the greedy method, we use the higher order power method [12], its symmetric version, and the bisymmetric power method [25] to compute the best rank-one approximation of each of the three kinds of tensors, respectively. Since all the best rank-one approximation problems for higher order tensors are NP-hard, the solution found by the power method is only an approximate value of the best rank-one approximation.…”

Conditions for strong ellipticity and M-eigenvalues

“…[12,27]. If λ ∈ R, x ∈ R m and y ∈ R n satisfy (6), we call λ an M-eigenvalue of A , and call x and y left and right M-eigenvectors of A , associated with the M-eigenvalue λ.…”

<i>H</i>-Singular Value of a Positive Tensor

“…Recently, eigenvalue problems for tensors have gained special attention in the realm of numerical multilinear algebra [1]- [4], and they have a wide range of practical applications [5] [6]. The definition of eigenvalues of square tensors has been introduced in [7]- [9].…”