A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible M-matrix

Abstract

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.