Advertisement

The Calculation of Specified Eigenvectors by Inverse Iteration

  • G. Peters
  • J. H. Wilkinson
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 186)

Abstract

When an approximation μ is known to an eigenvalue of a matrix A, inverse iteration provides an efficient algorithm for computing the corresponding eigenvector. It consists essentially of the determination of a sequence of vectors x r defined by
$$ \matrix{ {(A - \mu I){x_{r + 1}} = {k_r}{x_r}} & {(r = 0,1, \ldots ),} \cr } $$
(1)
where k r is chosen so that ‖x r+1‖ = l in some norm and x 0 is an arbitrary unit vector.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Barth, W., Martin, R. S., Wilkinson, J.H.: Calculation of the eigenvalues of a symmetric tridiagonal matrix by the method of bisection. Numer. Math. 9, 386–393 (1967). Cf. II/5.MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bowdler, Hilary, Martin, R. S., Reinsch, C, Wilkinson, J. H.: The QR and QL algorithms for symmetric matrices. Numer. Math. 11, 293–306 (1968). Cf. II/3.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Martin, R. S., Peters, G., Wilkinson, J. H.: The QR algorithm for real Hessenberg matrices. Numer. Math. 14, 219–231 (1970). Cf. 11/14.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Martin, R. S., Peters, G., Wilkinson, J. H. Reinsch, C, Wilkinson, J. H.: Householder’s tridiagonalization of a symmetric matrix. Numer. Math. 11, 181–195 (1968). Cf. II/2.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Martin, R. S., Peters, G., Wilkinson, J. H. Wilkinson, J.H.: Similarity reduction of a general matrix to Hessenberg form. Numer. Math. 12, 349–368 (1968). Cf.II/13.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Martin, R. S., Peters, G., Wilkinson, J. H.The modified LR algorithm for complex Hessenberg matrices. Numer. Math. 12, 369–376 (1968). Cf. II/16.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Martin, R. S., Peters, G., Wilkinson, J. H.The implicit QL algorithm. Numer. Math. 12, 377–383 (1968). Cf. II/4.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Peters, G., Wilkinson, J. H.: Eigenvectors of real and complex matrices by LR and QR triangularizations. Numer. Math. 16, 181–204 (1970). Cf. II/15.MathSciNetCrossRefGoogle Scholar
  9. 9.
    Peters, G., Wilkinson, J. HTheoretical and practical properties of inverse iteration. (To be published.)Google Scholar
  10. 10.
    Varah, J.: Ph. D. Thesis. Stanford University (1967).Google Scholar

Copyright information

© Springer-Verlag Berlin · Heidelberg 1971

Authors and Affiliations

  • G. Peters
  • J. H. Wilkinson

There are no affiliations available

Personalised recommendations