Journal of Mathematical Sciences

, Volume 168, Issue 3, pp 417–419 | Cite as

Root-squaring with DPR1 matrices

  • V. Y. Pan

Recent progress in polynomial root-finding relies on employing the associated companion and generalized companion DPR1 matrices. (“DPR1” stands for “diagonal plus rank-one.”) We propose an algorithm that uses nearly linear arithmetic time to square a DPR1 matrix. Consequently, the algorithm squares the roots of the associated characteristic polynomial. This incorporates the classical techniques of polynomial root-finding by means of root-squaring into a new effective framework. Our approach is distinct from the earlier fast methods for squaring companion matrices. Bibliography: 13 titles.


Recent Progress Characteristic Polynomial Fast Method Generalize Companion Classical Technique 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    D. A. Bini, L. Gemignani, and V. Y. Pan, “Inverse power and Durand-Kerner iteration for univariate polynomial root-finding,” Comput. Math. Appl., 47, No. 2–3, 447–459 (2004).MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    D. A. Bini, L.Gemignani, and V. Y. Pan, “Improved initialization of the accelerated and robust QR-like polynomial root-finding,” Electron. Trans. Numer. Anal., 17, 195–205 (2004).MATHMathSciNetGoogle Scholar
  3. 3.
    D. A. Bini, L. Gemignani, and V. Y. Pan, “Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equation,” Numer. Math., 100, No. 3, 373–408 (2005).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    J. P. Cardinal, “On two iterative methods for approximating the roots of a polynomial,” in: J. Renegar, M. Shub, and S. Smale (eds.), Proceedings of the AMS-SIAM Summer Seminar “Mathematics of Numerical Analysis: Real Number Algorithms,” Lect. Appl. Math., 32, Amer. Math. Soc., Providence, Rhode Island (1996), pp. 165–188.Google Scholar
  5. 5.
    S. Fortune, “An iterated eigenvalue algorithm for approximating roots of univariate polynomials,” J. Symbolic Comput., 33, No. 5, 627–646 (2002).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    A. S. Householder, “Dandelin, Lobatchevskii, or Gräffe?”, Amer. Math. Monthly, 66, 464–466 (1959).MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    A. S. Householder, The Theory of Matrices in Numerical Analysis, Dover, New York (1964).MATHGoogle Scholar
  8. 8.
    F. Malek and R. Vaillanourt, “Polynomial zero finding iterative matrix algorithms,” Comput. Math. Appl., 29, No. 1, 1–13 (1995).MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    F. Malek and R. Vaillanourt, “A composite polynomial zero finding matrix algorithm,” Comput. Math. Appl., 30, No. 2, 37–47 (1995).MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. Y. Pan, Structured Matrices and Polynomials: Unifed Superfast Algorithms, Birkhäuser/Springer, Boston-New York (2001).Google Scholar
  11. 11.
    V. Y. Pan, “Univariate polynomials: nearly optimal algorithms for factorization and root finding,” J. Symbolic Comput., 33, No. 5, 701–733 (2002).MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    V. Y. Pan, “Amended DseSC power method for polynomial root-finding,” Comput. Math. Appl., 49, No. 9–10, 1515–1524 (2005).MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    V. Y. Pan, D. Ivolgin, B. Murphy, R. E. Rosholt, Y. Tang, X. Wang, and X. Yan, “Root-finding with eigen-solving,” in: Dongming Wang and Li-Hong Zhi (eds.), Symbolic-Numeric Computation, Birkhäuser, Basel-Boston (2007), Chap.14, pp. 219–245.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2010

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceLehman College of the City University of New YorkBronxUSA

Personalised recommendations