Linear homotopy method for computing generalized tensor eigenpairs

Abstract

Let m, m^{\prime }, n be positive integers such that m\ne m^{\prime }. Let A be an mth order n-dimensional tensor, and let B be an m^{\prime }th order n-dimensional tensor. λ ∈ ℂ is called a B-eigenvalue of A if A{x}^{m-\mathrm{1}}=\lambda B{x}^{m^{\prime }-\mathrm{1}} and B{x}^{m^{\prime }}=\mathrm{1} for some x ∈{ℂ}^n\backslash \{\mathrm{0}\}. In this paper, we propose a linear homotopy method for solving this eigenproblem. We prove that the method finds all isolated B-eigenpairs. Moreover, it is easy to implement. Numerical results are provided to show the efficiency of the proposed method.

Keywords

Tensors / generalized eigenpairs / polynomial systems / linear homotopy

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Higher Education Press and Springer-Verlag GmbH Germany