Skip to main content
Springer Nature Link
Log in

A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible \(M\)-matrix

  • Published:
Numerische Mathematik Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

In this paper, based on the Noda iteration, we present inexact Noda iterations (INI), to find the smallest eigenvalue and the associated positive eigenvector of a large irreducible nonsingular \(M\)-matrix. The positivity of approximations is critical in applications, and if the approximations lose the positivity then they may be meaningless and could not be interpreted. We propose two different inner tolerance strategies for solving the inner linear systems involved, and prove that the convergence of resulting INI algorithms is globally linear and superlinear with the convergence order \(\frac{1+\sqrt{5}}{2}\), respectively. The proposed INI algorithms are structure preserving and maintains the positivity of approximate eigenvectors. We also revisit the exact Noda iteration and establish a new quadratic convergence result. All the above is first done for the problem of computing the Perron root and the positive Perron vector of an irreducible nonnegative matrix and is then adapted to computing the smallest eigenpair of the irreducible nonsingular \(M\)-matrix. Numerical examples illustrate that the proposed INI algorithms are practical, and they always preserve the positivity of approximate eigenvectors. We compare them with the Jacobi–Davidson method, the implicitly restarted Arnoldi method and the explicitly restarted Krylov–Schur method, all of which cannot guarantee the positivity of approximate eigenvectors, and illustrate that the overall efficiency of the INI algorithms is competitive with and can be considerably higher than the latter three methods.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Notes

  1. We thank one of the referees, who suggested to us this more direct proof than our original one.

References

  1. Abate, J., Choudhury, L., Whitt, W.: Asymptotics for steady-state tail probabilities in structured Markov queueing models. Commun. Stat. Stoch. Models 1, 99–143 (1994)

    Article  MathSciNet  Google Scholar 

  2. AIM@SHAPE: Advanced and innovative models and tools for the development of semantic-based systems for handling, acquiring, and processing knowledge embedded in multidimensional digital objects. http://shapes.aim-at-shape.net

  3. Alfa, A.S., Xue, J., Ye, Q.: Accurate computation of the smallest eigenvalue of a diagonally dominant \(M\)-matrix. Math. Comput. 71, 217–236 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Berman, A., Plemmons, R.J.: Nonnegative matrices in the mathematical sciences. In: Classics in Applied Mathematics, Vol. 9. SIAM, Philadelphia (1994)

  5. Berns-Müller, J., Graham, I.G., Spence, A.: Inexact inverse iteration for symmetric matrices. Linear Algebra Appl. 416, 389–413 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Collatz, L.: Einchliessungenssatz für die characteristischen Zahlen von Matrizen. Math. Z. 48, 221–226 (1942)

    Article  MathSciNet  Google Scholar 

  7. DIMACS10: DIMACS10 test set and the University of Florida Sparse Matrix Collection. http://www.cise.ufl.edu/research/sparse/dimacs10/

  8. Elsner, L.: Inverse iteration for calculating the spectral radius of a nonnegative irreducible matrix. Linear Algebra Appl. 15, 235–242 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  9. Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. The John Hopkins University Press, Baltimore (2012)

    Google Scholar 

  10. Gu, X.D., Huang, T.-M., Huang, W.-Q., Lin, S.-S., Lin, W.-W., Yau, S.-T.: High performance computing for spherical conformal and Riemann mappings. In: Technical Report, Department of Applied Mathematics, National Chiao Tung University, Taiwan (2013)

  11. Hernández, V., Román, J.E., Tomás, A., Vidal, V.: Krylov–Schur methods in SLEPc. In: SLEPc Technical Report STR-7 (2007). http://www.grycap.upv.es/slepc

  12. Horn, R.A., Johnson, C.R.: Matrix Analysis. The Cambridge University Press, Cambridge (1985)

    Book  MATH  Google Scholar 

  13. Jia, Z.: On convergence of the inexact Rayleigh quotient iteration with MINRES. J. Comput. Appl. Math. 236, 4276–4295 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Jia, Z.: On convergence of the inexact Rayleigh quotient iteration with the Lanczos method used for solving linear systems. Sci. China Math. 56, 2145–2160 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  15. Jia, Z., Li, C.: On inner iterations in the shift-invert residual Arnoldi method and the Jacobi–Davidson method. Sci. China Math. 57, 1733–1752 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  16. Langville, A., Meyer, C.: Google’s PageRank and Beyond: The Science of Search Engine Rankings. The Princeton University Press, Princeton (2006)

    Book  Google Scholar 

  17. Lee, C.: Residual Arnoldi method: theory, package and experiments. Ph.D. thesis, TR-4515, Department of Computer Science, University of Maryland at College Park (2007)

  18. Lai, Y.-L., Lin, K.-Y., Lin, W.-W.: An inexact inverse iteration for large sparse eigenvalue problems. Numer. Linear Algebra Appl. 4, 425–437 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu, C.-S.: Inexact iterative methods for solving eigenvalue problems. Ph.D. thesis, Department of Mathematics, National Tsinghua University, Hsinchu (2012)

  20. Lynn, M.S., Timlake, W.P.: Bounds for Perron eigenvectors and subdominant eigenvalues of positive matrices. Linear Algebra Appl. 2, 143–152 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  21. Noda, T.: Note on the computation of the maximal eigenvalue of a non-negative irreducible matrix. Numer. Math. 17, 382–386 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  22. Notay, Y.: JDCG and JDRPCG codes. http://homepages.ulb.ac.be/ynotay/

  23. Ostrowski, A.M.: On the convergence of the Rayleigh quotient iteration for the computation of the characteristic roots and vectors. V. (Usual Rayleigh quotient for non-Hermitian matrices and linear elementary divisors). Arch. Ration. Mech. Anal. 3, 472–481 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  24. Parlett, B.N.: The symmetric eigenvalue problem. In: Classics in Applied Mathematics, Vol. 20. SIAM, Philadelphia (1998)

  25. Robbé, M., Sadkane, M., Spence, A.: Inexact inverse subspace iteration with preconditioning applied to non-Hermitian eigenvalue problems. SIAM J. Matrix Anal. Appl. 31, 92–113 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Robertazzi, G.: Computer Networks and Systems: Queueing Theory and Performance Evaluation. Springer-Verlag, New York (1990)

  27. Saad, Y.: Numerical methods for large eigenvalue problems, revised version. In: Classics in Applied Mathematics, Vol. 66. SIAM, Philadelphia (2011)

  28. Sorensen, D.C.: Implicit application of polynomial filters in a \(k\)-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357–385 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  29. Shivakumar, P., Williams, J., Ye, Q., Marinov, C.: On two-sided bounds related to weakly diagonally dominant \(M\)-matrices with applications to digital circuit dynamics. SIAM J. Matrix Anal. Appl. 17, 298–312 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  30. Simoncini, V., Elden, L.: Inexact Rayleigh quotient-type methods for eigenvalue computations. BIT 42, 159–182 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sleijpen, G.L.G., van der Vorst, H.A.: A Jacobi–Davidson iteration method for linear eigenvalue problems. SIAM J. Matrix Anal. Appl. 17, 401–425 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sleijpen, G.L.G.: JDQR and JDQZ codes. http://www.math.uu.nl/people/sleijpen

  33. Stewart, G.W.: Matrix Algorithms, vol. II. SIAM, Philadelphia (2001)

    Book  MATH  Google Scholar 

  34. Stewart, G.W.: A Krylov–Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23, 601–614 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  35. Varga, R.S.: Matrix Iterative Analysis, 2nd edn. Spinger-Verlag, Berlin (2000)

    Book  MATH  Google Scholar 

  36. Wei, M.Q., Pang, M.Y., Fan, C.L.: Survey on planar parameterization of triangular meshes. In: Proceedings of the 2010 International Conference on Measuring Technology and Mechatronics Automation, pp. 702–705 (2010)

  37. Wielandt, H.: Unzerelegbare, nicht negativen Matrizen. Math. Z. 52, 642–648 (1950)

    Article  MATH  MathSciNet  Google Scholar 

  38. Wilkinson, J.H.: The Algebraic Eigenvalue Problem. Clarendon Press, Oxford (1965)

    MATH  Google Scholar 

  39. Xue, F., Szyld, D.B.: Efficient preconditioned inner solves for inexact Rayleigh quotient iteration and their connections to the single-vector Jacobi–Davidson method. SIAM J. Matrix Anal. Appl. 32, 993–1018 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  40. Xue, J.: Computing the smallest eigenvalue of an \(M\)-matrix. SIAM J. Matrix Anal. Appl. 17, 748–762 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We thank the referees for their careful reading of our paper and a number of valuable comments, which helped us improve the presentation of the paper.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China

    Zhongxiao Jia

  2. Department of Applied Mathematics, National Chiao Tung University, Hsinchu, 300, Taiwan

    Wen-Wei Lin & Ching-Sung Liu

Authors
  1. Zhongxiao Jia
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Wen-Wei Lin
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Ching-Sung Liu
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Zhongxiao Jia.

Additional information

Z. Jia was supported in part by the National Basic Research Program of China 2011CB302400 and the National Science Foundation of China (No. 11371219), and W.-W. Lin and C.-S. Liu were supported in part by the National Science Council, the National Center for Theoretical Sciences, the Center of MMSC, and ST Yau Center at Chiao-Da in Taiwan.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jia, Z., Lin, WW. & Liu, CS. A positivity preserving inexact Noda iteration for computing the smallest eigenpair of a large irreducible \(M\)-matrix. Numer. Math. 130, 645–679 (2015). https://doi.org/10.1007/s00211-014-0677-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00211-014-0677-2

Mathematics Subject Classification