A tensor is a multidimensional array. In general, an $m$th-order and $n$-dimensional tensor can be indexed as $\cA = (A_{i_1i_2\dots i_m})$, where $A_{i_1i_2\dots i_m} \in \bC$ for $1 \le i_1, i_2, \dots, i_m \le n$. Let ${\mathcal A}$ be an $m$th order $n$-dimensional tensor and ${\mathcal B}$ be an $m'$th order $n$-dimensional tensor. A scalar $\lambda \in {\mathbb C} $ and a vector $x \in {\mathbb C}^n \backslash \{0\}$ is called a generalized ${\mathcal B}$-eigenpair of ${\mathcal A}$ if ${\mathcal A} x^{m-1} = \lambda {\mathcal B} x^{m'-1}$ with ${\mathcal B}x^{m'} =1$ when $m \ne m'$. Different choices of $\cB$ 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 $m \ne m'$. With proper implementation, these methods can find all equivalence classes of isolated eigenpairs. A MATLAB software package \texttt{TenEig 2.0} has been developed to implement these methods. Numerical results are provided to show its efficiency and effectiveness.
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