  In this paper, based on the Noda iteration, we present inexact Noda
iterations (INI), to find the smallest eigenvalue and the associated positive
eigenvector of a large irreducible nonsingular M-matrix. The positivity of
approximations is critical in applications, and if the approximations lose the
positivity then they will be physically meaningless. We propose two different
inner tolerance strategies for solving the inner linear systems involved, and
prove that the resulting INI algorithms are globally linear and superlinear
with the convergence order $\frac{1+\sqrt{5}}{2}$, respectively. The proposed
INI algorithms are structure preserving and maintains the positivity of
approximate eigenvectors. We also revisit the exact Noda iteration and
establish a new quadratic convergence result. All the above is first done for
the problem of computing the Perron root and the positive Perron vector of an
irreducible nonnegative matrix and then adapted to that of computing the
smallest eigenpair of the irreducible nonsingular M-matrix. Numerical examples
illustrate that the proposed INI algorithms are practical, and they always
preserve the positivity of approximate eigenvectors. We compare them with the
positivity non-preserving Jacobi--Davidson method and implicitly restarted
Arnoldi method, which often compute physically meaningless eigenvectors, and
illustrate that the overall efficiency of the INI algorithms is competitive
with and can be considerably higher than the latter two methods.
