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

An inexact Noda iteration for computing the smallest eigenpair of a large irreducible monotone matrix