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.

The standard approaches to solving an overdetermined linear system Ax ≈ b find minimal corrections to the vector b and/or the matrix A such that the corrected system is consistent, such as the least squares (LS), the data least squares (DLS) and the total least squares (TLS). The scaled total least squares (STLS) method unifies the LS, DLS and TLS methods. The classical normwise condition numbers for the LS problem have been widely studied.However, there are no such similar results for the TLS and the STLS problems. In this paper, we first present a perturbation analysis of the STLS problem, which is a generalization of the TLS problem, and give a normwise condition number for the STLS problem. Different from normwise condition numbers, which measure the sizes of both input perturbations and output errors using some norms, componentwise condition numbers take into account the relation of each data component, and possible data sparsity. Then in this paper we give explicit expressions for the estimates of the mixed and componentwise condition numbers for the STLS problem. Since the TLS problem is a special case of the STLS problem, the condition numbers for the TLS problem follow immediately from our STLS results. All the discussions in this paper are under the Golub-Van Loan condition for the existence and uniqueness of the STLS solution.

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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.

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