Numerical Algorithms

, Volume 80, Issue 3, pp 937–955

# Noda iterations for generalized eigenproblems following Perron-Frobenius theory

Original Paper

## Abstract

In this paper, we investigate the generalized eigenvalue problem Ax = λBx arising from economic models. Under certain conditions, there is a simple generalized eigenvalue ρ(A, B) in the interval (0, 1) with a positive eigenvector. Based on the Noda iteration, a modified Noda iteration (MNI) and a generalized Noda iteration (GNI) are proposed for finding the generalized eigenvalue ρ(A, B) and the associated unit positive eigenvector. It is proved that the GNI method always converges and has a quadratic asymptotic convergence rate. So GNI has a similar convergence behavior as MNI. The efficiency of these algorithms is illustrated by numerical examples.

## Keywords

Generalized eigenproblem Generalized Noda iteration Nonnegative irreducible matrix M-matrix Quadratic convergence Perron-Frobenius theory

65F15 65F99

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## Authors and Affiliations

• Xiao Shan Chen
• 1
Email author
• Seak-Weng Vong
• 2
• Wen Li
• 1
• Hongguo Xu
• 3
1. 1.School of Mathematical SciencesSouth China Normal UniversityGuangzhouPeople’s Republic of China
2. 2.Department of MathematicsUniversity of MacauMacauChina
3. 3.Department of MathematicsUniversity of KansasLawrenceUSA