| A tensor is a multidimensional array. In general, an mth-order and n-dimensional tensor can be indexed as A = (Ai1i 2...im), where AAi1i 2...im ∈ C for 1 ≤ i1,i 2;...; im ≤ n. Let A be an mth order n -dimensional tensor and B be an m'th order n-dimensional tensor. A scalar gamma ∈ C and a vector x ∈ Cn{0} is called a generalized B-eigenpair of A if A xm-1 = gammaB xm'-1 with Bxm' = 1 when m ≠ m'. Different choices of B yield different versions of the tensor eigenvalue problem.;As one can see, computing tensor eigenpairs amounts to solving a polynomial system. To find all solutions of a polynomial system, the homotopy continuation methods are very useful in terms of computational cost and storage space. By taking advantage of the solution structure of the tensor eigenproblem, two easy-to-implement linear homotopy methods which follow the optimal number of paths will be proposed to solve the generalized tensor eigenproblem when m ≠ m'. With proper implementation, these methods can find all equivalence classes of isolated eigenpairs. A MATLAB software package TenEig 2.0 has been developed to implement these methods. Numerical results are provided to show its efficiency and effectiveness. |
