Skip to main content

Computation of the Spectra and Eigenvectors of Large Complex Matrices

  • Chapter
  • First Online:
Selberg Zeta Functions and Transfer Operators

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2139))

  • 1092 Accesses

Abstract

One of the critical points in our numerical investigation of the transfer operator is the computation of its eigenvalues. In this section we want to describe briefly what problems arise when computing the eigenvalues of the transfer operator and how we can overcome these problems. To get the best results, both with respect to accuracy and computation time, we had to combine several techniques to produce an optimal algorithm.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Bindel, D., Demmel, J., Kahan, W., Marques, O.: On computing Givens rotations reliably and efficiently. ACM Trans. Math. Softw. 28, 206–238 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  2. Braman, K., Byers, R., Mathias, R.: The multishift QR algorithm. Part I and II: maintaining well-focused shifts and level 3 performance and aggressive early deflation. SIAM J. Matrix Anal. Appl. 23 (4), 929–973 (2002)

    Google Scholar 

  3. Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33 (2), 13:1–13:15 (2007)

    Google Scholar 

  4. Francis, J.: The QR Transformation A Unitary Analogue to the LR Transformation Part 1. Comput. J. 4 (3), 265–271 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  5. Francis, J.: The QR Transformation Part 2. Comput. J. 4 (4), 332–345 (1962)

    Article  MathSciNet  Google Scholar 

  6. Givens, W.: Computation of plane unitary rotations transforming a general matrix to triangular form. J. Soc. Ind. Appl. Math. 6 (1), 26–50 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  7. Golub, G., Loan, C.V.: Matrix Computations. Johns Hopkins University Press, Baltimore (1996)

    MATH  Google Scholar 

  8. Schwarz, H.: Numerische Mathematik. B.G. Teubner Stuttgart (1997)

    Google Scholar 

  9. Watkins, D.: The QR algorithm revisited. SIAM Rev. 50 (1), 133–145 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Fraczek, M.S. (2017). Computation of the Spectra and Eigenvectors of Large Complex Matrices. In: Selberg Zeta Functions and Transfer Operators. Lecture Notes in Mathematics, vol 2139. Springer, Cham. https://doi.org/10.1007/978-3-319-51296-9_5

Download citation

Publish with us

Policies and ethics