  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.
