  In this paper, we present an inexact Noda iteration with inner-outer
iterations for finding the smallest eigenvalue and the associated eigenvector
of an irreducible monotone matrix. The proposed inexact Noda iteration contains
two main relaxation steps for computing the smallest eigenvalue and the
associated eigenvector, respectively. These relaxation steps depend on the
relaxation factors, and we analyze how the relaxation factors in the relaxation
steps affect the convergence of the outer iterations. By considering two
different relaxation factors for solving the inner linear systems involved, we
prove that the convergence is globally linear or superlinear, depending on the
relaxation factor, and that the relaxation factor also influences the
convergence rate. The proposed inexact Noda iterations are structure preserving
and maintain the positivity of approximate eigenvectors. Numerical examples are
provided to illustrate that the proposed inexact Noda iterations are practical,
and they always preserve the positivity of approximate eigenvectors.
