In this paper, we propose the tensor Noda iteration (NI) and its inexact version for solving the eigenvalue problem of a particular class of tensor pairs called generalized $\mathcal{M}$-tensor pairs. A generalized $\mathcal{M}$-tensor pair consists of a weakly irreducible nonnegative tensor and a nonsingular $\mathcal{M}$-tensor within a linear combination. It is shown that any generalized $\mathcal{M}$-tensor pair admits a unique positive generalized eigenvalue with a positive eigenvector. A modified tensor Noda iteration(MTNI) is developed for extending the Noda iteration for nonnegative matrix eigenproblems. In addition, the inexact generalized tensor Noda iteration method (IGTNI) and the generalized Newton-Noda iteration method (GNNI) are also introduced for more efficient implementations and faster convergence. Under a mild assumption on the initial values, the convergence of these algorithms is guaranteed. The efficiency of these algorithms is illustrated by numerical experiments.

翻译：本文提出用于求解一类特定张量对（称为广义$\mathcal{M}$-张量对）特征值问题的张量Noda迭代（NI）及其非精确版本。广义$\mathcal{M}$-张量对由线性组合中的弱不可约非负张量与非奇异$\mathcal{M}$-张量构成。研究表明，任意广义$\mathcal{M}$-张量对均存在唯一的正广义特征值及对应的正特征向量。为将Noda迭代推广至非负矩阵特征问题，我们发展了修正张量Noda迭代（MTNI）。此外，为提升计算效率与收敛速度，还引入了非精确广义张量Noda迭代方法（IGTNI）与广义牛顿-Noda迭代方法（GNNI）。在初始值的温和假设下，这些算法的收敛性得到保证。数值实验验证了算法的有效性。