Advertisement

Real and Complex Polynomial Root-Finding by Means of Eigen-Solving

  • Victor Y. Pan
  • Guoliang Qian
  • Ai-Long Zheng
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)

Abstract

Our new numerical algorithms approximate real and complex roots of a univariate polynomial lying near a selected point of the complex plane, all its real roots, and all its roots lying in a fixed half-plane or in a fixed rectangular region. The algorithms seek the roots of a polynomial as the eigenvalues of the associated companion matrix. Our analysis and experiments show their efficiency. We employ some advanced machinery available for matrix eigen-solving, exploit the structure of the companion matrix, and apply randomized matrix algorithms, repeated squaring, matrix sign iteration and subdivision of the complex plane. Some of our techniques can be of independent interest.

Keywords

Root-finding Eigen-solving Randomization Matrix sign function 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bini, D.A., Gemignani, L., Pan, V.Y.: Inverse power and Durand/Kerner iteration for univariate polynomial root-finding. Computers and Math (with Applics.) 47(2/3), 447–459 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Bini, D., Pan, V.Y.: Graeffe’s, Chebyshev, and Cardinal’s processes for splitting a polynomial into factors. J. Complexity 12, 492–511 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Cardinal, J.P.: On two iterative methods for approximating the roots of a polynomial. Lectures in Applied Math 32, 165–188 (1996)MathSciNetGoogle Scholar
  4. 4.
    Emiris, I.Z., Mourrain, B., Tsigaridas, E.P.: Real Algebraic Numbers: Complexity Analysis and Experimentation. In: Hertling, P., Hoffmann, C.M., Luther, W., Revol, N. (eds.) Real Number Algorithms. LNCS, vol. 5045, pp. 57–82. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Golub, G.H., Van Loan, C.F.: Matrix Computations, 3rd edn. Johns Hopkins University Press, Baltimore (1996)zbMATHGoogle Scholar
  6. 6.
    Higham, N.J.: Functions of Matrices: Theory and Computations. SIAM (2008)Google Scholar
  7. 7.
    Pan, C.–T.: On the existence and computation of Rrank-revealing LU factorization. Linear Algebra and Its Applications 316, 199–222 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser, Boston, and Springer, NY (2001)Google Scholar
  9. 9.
    Pan, V.Y.: Amended DSeSC power method for polynomial root-finding. Computers and Math (with Applics.) 49(9-10), 1515–1524 (2005)zbMATHCrossRefGoogle Scholar
  10. 10.
    Pan, V.Y., Qian, G., Murphy, B., Rosholt, R.E., Tang, Y.: Real root-finding. In: Vershelde, J., Stephen Watt, S. (eds.) Proc. Third Int. Workshop on Symbolic–Numeric Computation (SNC 2007), London, Ontario, Canada, pp. 161–169. ACM Press, New York (2007)Google Scholar
  11. 11.
    Pan, V.Y., Zheng, A.: New progress in real and complex polynomial root-finding. Computers and Math (Also in Proc. ISSAC 2010) 61, 1305–1334 (2010)MathSciNetGoogle Scholar
  12. 12.
    Pan, V.Y., Qian, G., Zheng, A.: Randomized Matrix Computations II. Tech. Report TR 2012006, Ph.D. Program in Computer Science, Graduate Center, the City University of New York (2012), http://www.cs.gc.cuny.edu/tr/techreport.php?id=433
  13. 13.
    Stewart, G.W.: Matrix Algorithms, Vol II: Eigensystems, 2nd edn. SIAM, Philadelphia (2001)zbMATHCrossRefGoogle Scholar
  14. 14.
    Yap, C., Sagraloff, M.: A simple but exact and efficient algorithm for complex root isolation. In: Proc. ISSAC 2011, pp. 353–360 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Victor Y. Pan
    • 1
    • 2
  • Guoliang Qian
    • 2
  • Ai-Long Zheng
    • 2
  1. 1.Department of Mathematics and Computer ScienceLehman College of the City University of New YorkBronxUSA
  2. 2.Ph.D. Programs in Mathematics and Computer ScienceThe Graduate Center of the City University of New YorkNew YorkUSA

Personalised recommendations