Skip to main content
SpringerLink
Log in

Matrix Eigenvector Methods

  • Chapter
Methods in Computational Molecular Physics

Part of the book series: NATO ASI Series ((ASIC,volume 113))

Abstract

Widespread use of the linear variation method in quantum mechanics leads to the need for efficient matrix eigenvalue methods for real symmetric matrices. Efficiency must be judged, however, in terms of the ability to solve a variety of eigenvalue problems on computers of widely different architecture, memory size, and data transfer rates. To further confuse the situation, cost rather than system throughput, is usually of paramount concern. Additionally, algorithms vary widely in ease of programming, simplicity, reliability, portability and availability as part of standard packages. Consequently there are a wide variety of eigenvalue algorithms in use, and a typical quantum laboratory will incorporate several of these into their program package.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover 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

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Parlett, B. N.: 1980, “The Symmetric Eigenvalue Problem”, Prentice-Hall, Englewood Cliffs, N.J.

    MATH  Google Scholar 

  2. Wilkinson J. H. and Reinsch, C.: 1971, “Handbook for Automatic Computation, Volume II—Linear Algebra”, Springer-Verlag, New York.

    Google Scholar 

  3. Corneil, D.: 1965, “Eigenvalues and Orthogonal Eigenvectors of Real Symmetric Matrices”, Masters Thesis, University of Toronto.

    Google Scholar 

  4. Birge, R. R. and Hubbard, L. M.: 1978, J. Comput. Phys. 29, pp. 199–207.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. Givens, W.: 1954, “Numerical Computation of the Characteristic Values of a Real Symmetric Matrix”, ORNL-1574, Oak Ridge National Laboratory.

    MATH  Google Scholar 

  6. Wilkinson, J. H.: 1960, Comp. J. 3, pp. 23–27.

    Article  MathSciNet  MATH  Google Scholar 

  7. Householder, A. S.: 1958, J. Soc. Ind. Appl. Math. 6, pp. 189–195.

    Article  MathSciNet  MATH  Google Scholar 

  8. Wilkinson, J. H.: 1965, “The Algebraic Eigenvalue Problem”, Oxford University Press, New York.

    MATH  Google Scholar 

  9. Moler, C. and Shavitt, I.: 1978, “Numerical Algorithms in Chemistry: Algebraic Methods”, LBL-8158 Lawrence Berkeley Laboratory.

    Google Scholar 

  10. Nesbet, R. K.: 1965, J. Chem. Phys. 43, pp. 311–312.

    Article  ADS  Google Scholar 

  11. Shavitt, I., Bender, C. F., Pipano, A., and Hosteny, R. P.: 1973, J. Comput. Phys. 11, pp. 90–108.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Shavitt, I.: 1970, J. Comput. Phys. 6, pp. 124–130.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. Shavitt, I.: 1977, unpublished report.

    Google Scholar 

  14. Raffenetti, R. C.: 1979, J. Comput. Phys. 32, pp. 403–419.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Lanczos, C.: 1950, J. Res. Natl. Bur. Stand. 45, pp. 255–282.

    MathSciNet  Google Scholar 

  16. Cullum, J. and Willoughby, R. A.: 1981, J. Comput. Phys. 44, pp. 329–358.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. Paige, C. C.: 1970, BIT 10, p. 183; 1972, J. Inst. Math. Applic. 10, pp. 373–381.

    Article  MathSciNet  ADS  Google Scholar 

  18. Cullum, J. and Donath, W. E.: 1974, “A Block Generalization of the Symmetric S-Step Lanczos Algorithm”, IBM Watson Research Center RC-4845; Saad, Y.: 1980, Soc. Ind. Appl. Math. J. Num. Anal. 17, p. 687.

    Google Scholar 

  19. Cullum, J. and Donath, W. E.: 1974, “A Block Generalization of the Symmetric S-Step Lanczos Algorithm”, IBM Watson Research Center RC-4845; Saad, Y.: 1980, Soc. Ind. Appl. Math. J. Num. Anal. 17, p. 687.

    MathSciNet  MATH  Google Scholar 

  20. Davidson, E. R.: 1975, J. Comput. Phys. 17, pp. 87–94.

    Article  ADS  MATH  Google Scholar 

  21. Kalamboukis, T. Z.: 1980, J. Phys. A: Math. Gen. 13, pp. 57–62.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Davidson. E. R.: 1980, J. Phys. A: Math. Gen. 13, pp. 179–180.

    Article  ADS  Google Scholar 

  23. Butscher, W. and Kammer, W. E.: 1976, J. Comput. Phys. 20, pp. 313–325.

    Article  MathSciNet  ADS  Google Scholar 

  24. Seeger, R. Krishman, R., and Pople, J. A.: 1978, J. Chem. Phys. 68, pp. 2519–2521.

    Article  ADS  Google Scholar 

  25. Weber, J., Lacroix, R., and Wanner, G.: 1980, Computers and Chemistry 4, pp. 55–60.

    Article  Google Scholar 

  26. Rettrup, S.: 1982, J. Comput. Phys. 45, pp. 100107.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Hirao, K., and Nakatsuji, H.: 1982, J. Comput. Phys. 45, pp. 246–254.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Cheung, L. M., and Bishop, D. M.: 1977, Computer Phys. Comm. 13, pp. 247–250.

    Article  ADS  Google Scholar 

  29. Gallup, G.: 1982, J. Comput. Chem. 3, pp. 127–129.

    Article  MathSciNet  Google Scholar 

  30. Ruhe, A.: 1975, J. Comput. Phys. 19, pp. 110–120.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Tsunematsu, T., and Takeda, T.: 1978, J. Comput. Phys. 28, pp. 287–293.

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chemistry Dept., University of Washington, USA

    Ernest R. Davidson

Authors
  1. Ernest R. Davidson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Max-Planck-Institute for Physics and Astrophysics, Garching, West Germany

    G. H. F. Diercksen

  2. Theoretical Chemistry Department, University of Oxford, Oxford, UK

    S. Wilson

Rights and permissions

Reprints and permissions

Copyright information

© 1983 D. Reidel Publishing Company, Dordrecht, Holland

About this chapter

Cite this chapter

Davidson, E.R. (1983). Matrix Eigenvector Methods. In: Diercksen, G.H.F., Wilson, S. (eds) Methods in Computational Molecular Physics. NATO ASI Series, vol 113. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7200-1_4

Download citation

  • DOI: https://doi.org/10.1007/978-94-009-7200-1_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-7202-5

  • Online ISBN: 978-94-009-7200-1

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation