A derivative-free spectral residual method for computing generalized eigenpairs of weakly symmetric tensors

This paper is concerned with computing generalized eigenpairs of weakly symmetric tensors. We first show that computing generalized eigenpairs of weakly symmetric tensors is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a derivative-free spectral residual method for it. The method utilizes the residual vector as the search direction and incorporates a derivative-free nonmonotone line search strategy, avoiding any explicit information associated with the Jacobian matrix of the considered nonlinear system of equations. Additionally, the global convergence of the method is established. Numerical results are presented to demonstrate superior efficiency of the proposed method through comparisons with the alternating least squares (ALS) method and other existing approaches. Furthermore, the results show that the proposed method can capture more, and in some cases all, generalized eigenvalues of weakly symmetric tensors.