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

TL;DR

This paper introduces an inexact Noda iteration method with relaxation steps for efficiently computing the smallest eigenpair of large irreducible monotone matrices, ensuring positivity preservation and analyzing convergence behavior.

Contribution

It proposes a novel inexact Noda iteration with relaxation factors, providing convergence analysis and demonstrating positivity preservation in eigenvector approximations.

Findings

Convergence is globally linear or superlinear depending on relaxation factors.

The method preserves the positivity of approximate eigenvectors.

Numerical examples confirm practical effectiveness and positivity preservation.

Abstract

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.