This paper is concerned with solving the multilinear systems \({\mathscr {A}}\mathbf {x}^{m-1}=\mathbf {b}\) whose coefficient tensors are mth-order and n-dimensional symmetric tensors. We first extend the accelerated overrelaxation (AOR) splitting method to solve the tensor equation. To improve the convergence, we develop a Newton-AOR (NAOR) method that hybridizes the Newton method and the accelerated overrelaxation scheme. Convergence analysis shows that the proposed methods converge under appropriate assumptions. Finally, some numerical examples are provided to show the effectiveness of the methods proposed.

中文翻译：

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求解对称张量方程的加速过松弛松弛方法

本文涉及求解系数张量为m阶且n维对称的多线性系统\（{\ mathscr {A}} \ mathbf {x} ^ {m-1} = \ mathbf {b} \）张量。我们首先扩展加速过松弛（AOR）分裂方法来求解张量方程。为了提高收敛性，我们开发了一种牛顿AOR（NAOR）方法，该方法将牛顿方法和加速过松弛方案混合在一起。收敛性分析表明，所提出的方法在适当的假设下收敛。最后，提供了一些数值例子来说明所提出方法的有效性。