Abstract

This work is a continuation of work by [J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, J. Matrix Anal. Appl., 36 (2015), pp. 942--973]. In that paper we introduced a companion QR algorithm that finds the roots of a polynomial by computing the eigenvalues of the companion matrix in $O(n^{2})$ time using $O(n)$ memory. We proved that the method is backward stable. Here we introduce, as an alternative, a companion QZ algorithm that solves a generalized eigenvalue problem for a companion pencil. More importantly, we provide an improved backward error analysis that takes advantage of the special structure of the problem. The improvement is also due, in part, to an improvement in the accuracy (in both theory and practice) of the turnover operation, which is the key component of our algorithms. We prove that for the companion QR algorithm, the backward error on the polynomial coefficients varies linearly with the norm of the polynomial's vector of coefficients. Thus, the companion QR algorithm has a smaller backward error than the unstructured QR algorithm (used by MATLAB's roots command, for example), for which the backward error on the polynomial coefficients grows quadratically with the norm of the coefficient vector. The companion QZ algorithm has the same favorable backward error as companion QR, provided that the polynomial coefficients are properly scaled.

Keywords

  1. polynomial
  2. root
  3. companion matrix
  4. companion pencil
  5. eigenvalue
  6. Francis algorithm
  7. QR algorithm
  8. QZ algorithm
  9. core transformation
  10. backward stability

MSC codes

  1. 65F15
  2. 65H17
  3. 15A18
  4. 65H04

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
J. Aurentz, R. Vandebril, and D. S. Watkins, Fast computation of the zeros of a polynomial via factorization of the companion matrix, SIAM J. Sci. Comput., 35 (2013), pp. A255--A269.
2.
J. Aurentz, R. Vandebril, and D. S. Watkins, Fast computation of eigenvalues of companion, comrade, and related matrices, BIT Numer. Math., 54 (2014), pp. 7--30.
3.
J. L. Aurentz, T. Mach, L. Robol, R. Vandebril, and D. S. Watkins, Fast and backward stable computation of roots of polynomials, part IIA: General backward error analysis, Technical report TW683, Department of Computer Science, University of Leuven, KU Leuven, Belgium, 2017.
4.
J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, Fast and backward stable computation of roots of polynomials, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 942--973.
5.
J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, Fast and stable unitary QR algorithm, Electron. Trans. Numer. Anal., 44 (2015), pp. 327--341.
6.
J. L. Aurentz, T. Mach, R. Vandebril, and D. S. Watkins, A note on companion pencils, Contemp. Math., 658 (2016), pp. 91--101.
7.
D. A. Bini, P. Boito, Y. Eidelman, L. Gemignani, and I. Gohberg, A fast implicit QR eigenvalue algorithm for companion matrices, Linear Algebra Appl., 432 (2010), pp. 2006--2031.
8.
D. A. Bini and L. Robol, Solving secular and polynomial equations: A multiprecision algorithm, J. Comput. Appl. Math., 272 (2014), pp. 276--292.
9.
P. Boito, Y. Eidelman, and L. Gemignani, Implicit QR for companion-like pencils, Math. Comp., 85 (2016), pp. 1753--1774.
10.
P. Boito, Y. Eidelman, and L. Gemignani, A Real QZ Algorithm for Structured Companion Pencils, preprint, arXiv:1608.05395, 2016.
11.
P. Boito, Y. Eidelman, L. Gemignani, and I. Gohberg, Implicit QR with compression, Indag. Math., 23 (2012), pp. 733--761.
12.
S. Chandrasekaran, M. Gu, J. Xia, and J. Zhu, A fast QR algorithm for companion matrices, Oper. Theory Adv. Appl., 179 (2007), pp. 111--143.
13.
F. De Terán, F. M. Dopico, and J. Pérez, Backward stability of polynomial root-finding using Fiedler companion matrices, IMA J. Numer. Anal., 36 (2016), pp. 133--173.
14.
B. Eastman, I.-J. Kim, B. Shader, and K. Vander Meulen, Companion matrix patterns, Linear Algebra Appl., 463 (2014), pp. 255--272.
15.
A. Edelman and H. Murakami, Polynomial roots from companion matrix eigenvalues, Math. Comp., 64 (1995), pp. 763--776.
16.
D. K. Faddeev and V. N. Faddeeva, Computational Methods of Linear Algebra, W. H. Freeman, New York, 1963.
17.
J. G. F. Francis, The QR transformation, part II, Comput. J., 4 (1961), pp. 332--345.
18.
F. R. Gantmacher, The Theory of Matrices, Vol. 1, Providence, RI, Chelsea, 1959.
19.
N. J. Higham, Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM, Philadelphia, PA, 2002.
20.
R. A. Horn and C. R. Johnson, Matrix Analysis, 2nd ed., Cambridge University Press, Cambridge, 2013.
21.
G. F. Jónsson and S. Vavasis, Solving polynomials with small leading coefficients, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 400--414.
22.
D. S. Mackey, N. Mackey, C. Mehl, and V. Mehrmann, Vector spaces of linearizations for matrix polynomials, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 971--1004.
23.
J. M. McNamee, A bibliography on roots of polynomials, J. Comput. Appl. Math., 47 (1993), pp. 391--394.
24.
C. B. Moler and G. W. Stewart, An algorithm for generalized matrix eigenvalue problems, SIAM J. Numer. Anal., 10 (1973), pp. 241--256.
25.
B. N. Parlett and C. Reinsch, Balancing a matrix for calculation of eigenvalues and eigenvectors, Numer. Math., 13 (1969), pp. 293--304.
26.
P. Van Dooren and P. Dewilde, The eigenstructure of an arbitrary polynomial matrix: Computational aspects, Linear Algebra Appl., 50 (1983), pp. 545--579.
27.
R. Vandebril and D. S. Watkins, An extension of the QZ algorithm beyond the Hessenberg-upper triangular pencil, Electron. Trans. Numer. Anal., 40 (2012), pp. 17--35.
28.
R. C. Ward, The combination shift QZ algorithm, SIAM J. Sci. Statist. Comput., 12 (1975), pp. 835--853.
29.
D. S. Watkins, Performance of the qz algorithm in the presence of infinite eigenvalues, SIAM J. Matrix Anal. Appl., 22 (2000), pp. 364--375.
30.
D. S. Watkins, The Matrix Eigenvalue Problem: GR and Krylov Subspace Methods, SIAM, Philadelphia, PA, 2007.
31.
D. S. Watkins, Fundamentals of Matrix Computations, 3rd ed., John Wiley & Sons, New York, 2010.

Information & Authors

Information

Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1245 - 1269
ISSN (online): 1095-7162

History

Submitted: 18 October 2017
Accepted: 25 May 2018
Published online: 14 August 2018

Keywords

  1. polynomial
  2. root
  3. companion matrix
  4. companion pencil
  5. eigenvalue
  6. Francis algorithm
  7. QR algorithm
  8. QZ algorithm
  9. core transformation
  10. backward stability

MSC codes

  1. 65F15
  2. 65H17
  3. 15A18
  4. 65H04

Authors

Affiliations

Funding Information

Onderzoeksraad, KU Leuven http://doi.org/10.13039/501100004497 : C4/16/056

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.