This paper is concerned with computing $𝒵$‐eigenpairs of symmetric tensors. We first show that computing $𝒵$‐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.

中文翻译：

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计算对称张量的𝒵本征对的局部和三次收敛算法

本文与计算有关 $𝒵$对称张量的本征对。我们首先证明计算$𝒵$对称张量的本征对等效于找到非线性方程组的非零解，然后为其提出改进的归一化牛顿法（MNNM）。我们提出的MNNM方法在某些合适的条件下被证明是局部和三次收敛的，这极大地改进了Jaffe，Weiss和Nadler最近提供的牛顿校正方法和正交牛顿校正方法，因为这两种方法仅具有二次收敛速度。作为一种应用，通过MNNM方法计算由量子纠缠计算产生的复值对称张量的unit对称本征对。数值结果表明了该方法的有效性。