Skip to main content
SpringerLink
Log in

The modified bordering method to evaluate eigenvalues and eigenvectors of normal matrices

  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

A bordering procedure is here proposed to evaluate the eigensystem of hermitian matrices, and more in general of normal matrices, when the spectral decomposition is known of then−1×n−1 principal minor. The procedure is also applicable to special real and nonsymmetric matrices here named quasi-symmetric. The computational cost to write the characteristic polynomial isO(n 2), using a new set of recursive formulas. A modified Brent algorithm is used to find the roots of the polynomial. The eigenvectors are evaluated in a direct way with a computational cost ofO(n 2) for each one. Some numerical considerations indicate where numerical difficulties may occur. Numerical results are given comparing this method with the Givens-Householder one.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A.C. Aitken,Determinanti e Matrici (Cremonese, Roma, 1967).

    Google Scholar 

  2. P. Arbenz and G.H. Gulub, QR-like algorithms for symmetric arrow matrices, SIAM J. Matrix Anal. Appl. 13 (1992) 655–658.

    Article  Google Scholar 

  3. R.P. Brent,Algorithms for Minimization without Derivatives (Prentice-Hall, 1973).

  4. C. Brezinski, Bordering methods and progressive forms for sequence transformations. Zastosow. Mat. 20 (1990) 435–443.

    Google Scholar 

  5. J.R. Bunch, C.P. Nielsen and D.C. Sorensen, Rank-one modification of the symmetric eigenproblem, Numer. Math. 31 (1978) 31–48.

    Article  Google Scholar 

  6. T.F. Chan and Y. Saad, Iterative methods for solving bordered systems with applications to continuation methods, SIAM J. Sci. Stat. Comput. 6 (1985) 438–451.

    Article  Google Scholar 

  7. J.J.M. Cuppen, A divide and conquer method for the symmetric tridiagonal eigenproblem, Numer. Math. 36 (1981) 177–195.

    Article  Google Scholar 

  8. V.N. Faddeeva,Computational Methods of Linear Algebra (Dover, New York, 1959).

    Google Scholar 

  9. G.E. Forsythe, Alternative derivations for Fox's escalator formulae for latent roots, Quart. J. Mech. Applied Math. 5 (1952) 191–195.

    Google Scholar 

  10. L. Fox, Escalator methods for latent roots, Quart. J. Mech. Applied Math. 5 (1952) 178–190.

    Google Scholar 

  11. J.N. Franklin,Matrix Theory (Prentice-Hall, Englewood Cliffs, 1968).

    Google Scholar 

  12. S.J. Hammarling,Latent Roots and Latent Vectors (Adam Hilger, London, 1970).

    Google Scholar 

  13. A.S. Householder,The Theory of Matrices in Numerical Analysis (Dover, New York, 1964).

    Google Scholar 

  14. N. Gastinel,Analyze Numérique Linéaire (Hermann, Paris, 1966).

    Google Scholar 

  15. G.H. Golub, Some modified matrix eigenvalue problems, SIAM Review 15 (1973) 318–334.

    Article  Google Scholar 

  16. G.H. Golub and C.F. Van Loan,Matrix Computation (The Johns Hopkins University Press, Baltimore and London, 1989).

    Google Scholar 

  17. R.T. Gregory and D.L. Karney,A Collection of Matrices for Testing Computational Algorithms (Wiley Interscience, New York, 1969).

    Google Scholar 

  18. J. Morris, An escalator process for the solution of linear simultaneous equations, Phil. Mag. 38 (1946) 106–110.

    Google Scholar 

  19. J. Morris and J.W. Head, The escalator process for the solution of Lagrangian frequency equations, Phil. Mag. 35 (1944) 735–759.

    Google Scholar 

  20. D.P. O'Leary and G. W. Stewart, Computing the eigenvalues and the eigenvectors of symmetric arrowhead matrices J. Comput. Phys. 90 (1990) 497–505.

    Article  MathSciNet  Google Scholar 

  21. B.N. Parlett,The Symmetric Eigenvalue Problem (Prentice-Hall, Englewood Cliffs, NJ., 1980).

    Google Scholar 

  22. J.B. Rosser, C. Lanczos, M.R. Hestenes and W. Karush, Separation of close eigenvalues of a real symmetric matrix. J. Res. Nat. Bur. Standards 47 (1951) 291–297.

    Google Scholar 

  23. B.T. Smith, J.M. Boyle, J.J. Dongarra, B.S. Garbow, Y. Ikebe, V.C. Klema and C.B. Moler,Matrix Eigensystem Routines — EISPACK Guide, Lecture Notes in Comp. Sci. vol. 6, 2nd ed. (Springer, Berlin, 1976).

    Google Scholar 

  24. D.C. Sorensen and P.T.P. Tang, On the orthogonality of eigenvectors computed by divide-and-conquer techniques. SIAM J. Numer. Anal. 28 (1991) 1752–1775.

    Google Scholar 

  25. W.F. Trench, Numerical solution of the eigenvalue problem for Hermitian Toeplitz matrices, SIAM J. Matrix Anal. Appl. 10 (1989) 135–146.

    Article  Google Scholar 

  26. B. Vinograde, Note on the escalator method, Proc. AMS 1 (1950) 162–164.

    Google Scholar 

  27. J.H. Wilkinson,The Algebraic Eigenvalue Problem (Clarendon Press, Oxford, England, 1965).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, Università di Padova, Via Belzoni 7, I-35131, Padova, Italy

    M. Morandi Cecchi

  2. Dipartimento di Matematica e Applicazioni, Università di Napoli “Federico II”, Via Cintia, Monte S. Angelo, I-80126, Napoli, Italy

    E. Di Nardo

Authors
  1. M. Morandi Cecchi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Di Nardo
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cecchi, M.M., Di Nardo, E. The modified bordering method to evaluate eigenvalues and eigenvectors of normal matrices. Numer Algor 11, 285–309 (1996). https://doi.org/10.1007/BF02142503

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02142503

Keywords

Search

Navigation