Numerical Algorithms

, Volume 80, Issue 3, pp 937–955 | Cite as

Noda iterations for generalized eigenproblems following Perron-Frobenius theory

  • Xiao Shan ChenEmail author
  • Seak-Weng Vong
  • Wen Li
  • Hongguo Xu
Original Paper


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.


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

Mathematics Subject Classification (2010)

65F15 65F99 


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We thank Prof. Weizhang Huang of University of Kansas for introducing his work [7] and providing the data of Example 5.3. We would like to thank an anonymous referee for his/her valuable comments.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

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

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