Noda iterations for generalized eigenproblems following Perron-Frobenius theory

  • Xiao Shan Chen
  • 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 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.


  1. 1.
    Bai, Z.J., Demmel, J., Dongarra, J., Ruhe, A., Van der Vorst, H.: Templates for the solution of algebraic eigenvalue problems: a practical guide. Philadelphia, SIAM (2011)MATHGoogle Scholar
  2. 2.
    Bapat, R.B., OLesky, D.D., van den Driesseche, P.: Perron-frobenius theory for a generalized eigenproblem. Linear Multi. Algebra 40, 141–152 (1995)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences, Classics Appl Math, vol. 9. SIAM, Philadelphia (1994)CrossRefGoogle Scholar
  4. 4.
    Elsner, L.: Inverse iteration for calculating the spectral radius of a non-negative irreducible matrix. Linear Algebra Appl. 15, 235–242 (1976)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fujimoto, T.: A generalization of the Frobenius theorem. Toyama University Econ. Rev. 23(2), 269–274 (1979)Google Scholar
  6. 6.
    Golub, G.H., Van Loan, C.F.: Matrix computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)Google Scholar
  7. 7.
    Huang, W.: Sign-preserving of principal eigenfunctions in p1 finite element approximation of eigenvalue problems of second-order elliptic operators. J. Comput. Phys. 274, 230–244 (2014)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Jia, Z.X., Lin, W.-W., Liu, C.S.: A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible M-matrix. Numer. Math. 130, 645–679 (2015)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Liu, C.-S.: An inexact Noda iteration for computing the smallest eigenpair of a large irreducible monotone matrix. Mathematics 130(4), 645–679 (2015)MATHGoogle Scholar
  10. 10.
    Liu, C.-S., Guo, C.-H., Lin, W.-W.: A positivity preserving inverse iteration for finding the perron pair of an irreducible nonnegative third tensor. SIAM J. Matrix Anal. Appl. 37, 911–932 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Morishima, M., Fujimoto, T.: The Frobenius theorem, its Solow Samuelson extension and the Kuhn-Tucker theorem. J. Math. Econ. 1, 199–205 (1974)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Noda, T.: Note on the computation of the maximal eigenvalue of a non-negative irreducible matrix. Numer. Math. 17, 382–386 (1971)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Pasinetti, L.L.: Essays on the theory of joint production. Macmillan Press, London (1980)CrossRefGoogle Scholar
  14. 14.
    Sraffa, P.: Production of commodities by means of commodities. Cambridge University Press, Cambridge (1960)Google Scholar
  15. 15.
    Stewart, G.W.: Matrix algorithms, Volume II: Eigensystems. SIAM, Philadelphia (2001)CrossRefMATHGoogle Scholar
  16. 16.
    Steward, G.W., Sun, J.G.: Matrix perturbation theory. Academic Press, Boston (1990)Google Scholar
  17. 17.
    Varga, R.S.: Matrix iterative analysis. Springer, Berlin (1962)Google Scholar
  18. 18.
    Xue, J.: Computing the smallest eigenvalue of an M-matrix. SIAM J. Matrix Anal. Appl. 17(4), 748–762 (1996)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Zhan, X.Z.: Matrix theory. American Mathematical Society Providence, Rhode Island (2013)CrossRefMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  • Xiao Shan Chen
    • 1
  • 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

Personalised recommendations