A locally and cubically convergent algorithm for computing Z-eigenpairs of symmetric tensors

This paper is concerned with computing -eigenpairs of symmetric tensors. We first show that computing -eigenpairs of a symmetric tensor is equivalent to finding the nonzero solutions of a nonlinear system of equations, and then propose a modified normalized Newton method (MNNM) for it. Our proposed MNNM method is proved to be locally and cubically convergent under some suitable conditions, which greatly improves the Newton correction method and the orthogonal Newton correction method recently provided by Jaffe, Weiss and Nadler since these two methods only enjoy a quadratic rate of convergence. As an application, the unitary symmetric eigenpairs of a complex-valued symmetric tensor arising from the computation of quantum entanglement in quantum physics are calculated by the MNNM method. Some numerical results are presented to illustrate the efficiency and effectiveness of our method.