Liangmin Zhou

Liping Chen, Lixing Han, Liangmin Zhou

We introduce the concept of mode-k generalized eigenvalues and eigenvectors of a tensor and prove some properties of such eigenpairs. In particular, we derive an upper bound for the number of equivalence classes of generalized tensor eigenpairs using mixed volume. Based on this bound and the structures of tensor eigenvalue problems, we propose two homotopy continuation type algorithms to solve tensor eigenproblems. With proper implementation, these methods can find all equivalence classes of isolated generalized eigenpairs and some generalized eigenpairs contained in the positive dimensional components (if there are any). We also introduce an algorithm that combines a heuristic approach and a Newton homotopy method to extract real generalized eigenpairs from the found complex generalized eigenpairs. A MATLAB software package TenEig has been developed to implement these methods. Numerical results are presented to illustrate the effectiveness and efficiency of TenEig for computing complex or real generalized eigenpairs.

Liping Chen, Lixing Han, Hongxia Yin et al.

In this paper we propose a homotopy method to compute the largest eigenvalue and a corresponding eigenvector of a nonnegative tensor. We prove that it converges to the desired eigenpair when the tensor is irreducible. We also implement the method using an prediction-correction approach for path following. Some numerical results are provided to illustrate the efficiency of the method.