Fast computation of eigenvalues of companion, comrade, and related matrices

Abstract

The class of eigenvalue problems for upper Hessenberg matrices of banded-plus-spike form includes companion and comrade matrices as special cases. For this class of matrices a factored form is developed in which the matrix is represented as a product of essentially 2×2 matrices and a banded upper-triangular matrix. A non-unitary analogue of Francis’s implicitly-shifted QR algorithm that preserves the factored form and consequently computes the eigenvalues in O(n 2) time and O(n) space is developed. Inexpensive a posteriori tests for stability and accuracy are performed as part of the algorithm. The results of numerical experiments are mixed but promising in certain areas. The single-shift version of the code applied to companion matrices is much faster than the nearest competitor.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    http://www.math.wsu.edu/students/jaurentz/publications/code.html.

References

  1. 1.

    Aurentz, J.L., Vandebril, R., Watkins, D.S.: Fast computation of the zeros of a polynomial via factorization of the companion matrix. SIAM J. Sci. Comput. 35, A255–A269 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Barnett, S.: Polynomials and Linear Control Systems. Dekker, New York (1983)

    Google Scholar 

  3. 3.

    Bini, D.A., Eidelman, Y., Gemignani, L., Gohberg, I.: Fast QR eigenvalue algorithms for Hessenberg matrices which are rank-one perturbations of unitary matrices. SIAM J. Matrix Anal. Appl. 29, 566–585 (2007)

    Article  MathSciNet  Google Scholar 

  4. 4.

    Bini, D.A., Boito, P., Eidelman, Y., Gemignani, L., Gohberg, I.: A fast implicit QR algorithm for companion matrices. Linear Algebra Appl. 432, 2006–2031 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Boito, P., Eidelman, Y., Gemignani, L., Gohberg, I.: Implicit QR with compression. Indag. Math. 23, 733–761 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Chandrasekaran, S., Gu, M., Xia, J., Zhu, J.: A fast QR algorithm for companion matrices. Oper. Theory, Adv. Appl. 179, 111–143 (2007)

    Article  MathSciNet  Google Scholar 

  7. 7.

    Eidelman, Y., Gemignani, L., Gohberg, I.: Efficient eigenvalue computation for quasiseparable Hermitian matrices under low rank perturbation. Numer. Algorithms 47, 253–273 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Fernando, K.V., Parlett, B.N.: Accurate singular values and differential qd algorithms. Numer. Math. 67, 191–229 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Fiedler, M.: A note on companion matrices. Linear Algebra Appl. 372, 325–331 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. 10.

    Francis, J.G.F.: The QR transformation, part II. Comput. J. 4, 332–345 (1962)

    Article  MathSciNet  Google Scholar 

  11. 11.

    Good, I.J.: The colleague matrix, a Chebyshev analogue of the companion matrix. Q. J. Math. 12, 61–68 (1961)

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

    Parlett, B.N.: The new qd algorithms. Acta Numer. 4, 459–491 (1995)

    Article  MathSciNet  Google Scholar 

  13. 13.

    Van Barel, M., Vandebril, R., Van Dooren, P., Frederix, K.: Implicit double shift QR-algorithm for companion matrices. Numer. Math. 116, 177–212 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Vandebril, R., Del Corso, G.M.: An implicit multishift QR-algorithm for Hermitian plus low rank matrices. SIAM J. Sci. Comput. 32, 2190–2212 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. 15.

    Vandebril, R., Van Barel, M., Mastronardi, N.: Matrix Computations and Semiseparable Matrices, Vol. II: Eigenvalue and Singular Value Methods. Johns Hopkins University Press, Baltimore (2008)

    Google Scholar 

  16. 16.

    Watkins, D.S.: The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods. SIAM, Philadelphia (2007)

    Google Scholar 

  17. 17.

    Watkins, D.S.: Fundamentals of Matrix Computations, 3rd edn. Wiley, New York (2010)

    Google Scholar 

  18. 18.

    Watkins, D.S.: Francis’s algorithm. Am. Math. Mon. 118(5), 387–403 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Zhlobich, P.: Differential qd algorithm with shifts for rank-structured matrices. SIAM J. Matrix Anal. Appl. 33, 1153–1171 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to David S. Watkins.

Additional information

The research was partially supported by the Research Council KU Leuven, projects OT/11/055 (Spectral Properties of Perturbed Normal Matrices and their Applications), CoE EF/05/006 Optimization in Engineering (OPTEC), by the Fund for Scientific Research–Flanders (Belgium) project G034212N (Reestablishing smoothness for matrix manifold optimization via resolution of singularities) and by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office, Belgian Network DYSCO (Dynamical Systems, Control, and Optimization).

Communicated by Peter Benner.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Aurentz, J.L., Vandebril, R. & Watkins, D.S. Fast computation of eigenvalues of companion, comrade, and related matrices. Bit Numer Math 54, 7–30 (2014). https://doi.org/10.1007/s10543-013-0449-x

Download citation

Keywords

  • Polynomial
  • Root
  • Companion matrix
  • Comrade matrix
  • LR algorithm

Mathematics Subject Classification

  • 65F15
  • 15A18