Affiliation:
1. Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung 811, Taiwan
Abstract
In this paper, we introduce an inexact Noda iteration method featuring inner and outer iterations for computing the smallest eigenvalue and corresponding eigenvector of an irreducible monotone matrix. The proposed method includes two primary relaxation steps designed to compute the smallest eigenvalue and its associated eigenvector. These steps are influenced by specific relaxation factors, and we examine how these factors impact the convergence of the outer iterations. By applying two distinct relaxation factors to solve the inner linear systems, we demonstrate that the convergence can be globally linear or superlinear, contingent upon the relaxation factor used. Additionally, the relaxation factor affects the rate of convergence. The inexact Noda iterations we propose are structure-preserving and ensure the positivity of the approximate eigenvectors. Numerical examples are provided to demonstrate the practicality of the proposed method, consistently preserving the positivity of approximate eigenvectors.
Funder
National Science and Technology Council
Reference25 articles.
1. Monotone positive stable matrices;Olesky;Linear Algebra Appl.,1983
2. Monotonicity and discretization error estimates;Axelsson;SIAM J. Numer. Anal.,1990
3. Axelsson, O., and Gololobov, S.V. (2003). Monotonicity and Discretization Error Estimates for Convection-Diffusion Problems, Department of Mathematics University of Nijmegen. Technical Report.
4. M-Matrices and generalizations using an operator theory approach;Schroeder;SIAM Rev.,1978
5. A new characterization of nonnegativity of Moore–Penrose inverses of Gram operators;Sivakumar;Positivity,2009