Advertisement

An effective implementation of a modified Laguerre method for the roots of a polynomial

  • Thomas R. CameronEmail author
Original Paper
First Online:
  • 34 Downloads

Abstract

Two common strategies for computing all roots of a polynomial with Laguerre’s method are explicit deflation and Maehly’s procedure. The former is only a semi-stable process and is not suitable for solving large degree polynomial equations. In contrast, the latter implicitly deflates the polynomial using previously accepted roots and is, therefore, a more practical strategy for solving large degree polynomial equations. However, since the roots of a polynomial are computed sequentially, this method cannot take advantage of parallel systems. In this article, we present an implementation of a modified Laguerre method for the simultaneous approximation of all roots of a polynomial. We provide a derivation of this method along with a detailed analysis of our algorithm’s initial estimates, stopping criterion, and stability. Finally, the results of several numerical experiments are provided to verify our analysis and the effectiveness of our algorithm.

Keywords

Laguerre’s method Polynomial roots Mathematical software 

Mathematics Subject Classification (2010)

26C10 65H04 65Y20 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

We are grateful to the NAG for providing a free trial that was used in the testing of our work. In addition, we are thankful for numerous private conversations with Dario Bini and David Watkins throughout our research and for the referees’ comments and suggestions that significantly improved this manuscript.

References

  1. 1.
    Aberth, O.: Iteration methods for finding all zeros of a polynomial simultaneously. Math. Comput. 27(122), 339–344 (1973)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Andrew, A.M.: Another efficient algorithm for convex hulls in two dimensions. Info. Proc. Lett. 9(15), 216–219 (1979)CrossRefGoogle Scholar
  3. 3.
    Aurentz, J.L., Mach, T., Vandebril, R., Watkins, D.S.: Fast and backward stable computation of roots of polynomials. SIAM J. Matrix Anal. Appl. 36(3), 942–973 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bini, D.A.: Numerical computation of polynomial zeros by means of Aberth’s method. Numer. Algor. 13, 179–200 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial root finder. Numer. Algor. 23, 127–173 (2000)CrossRefGoogle Scholar
  6. 6.
    Bini, D.A., Robol, L.: Solving secular and polynomial equations: a multiprecision algorithm. J. Comput. Appl. Math. 272, 276–292 (2014)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Börsch-Supan, W.: A posteriori error bounds for the zeros of polynomials. Numer. Math. 5, 380–398 (1963)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chaitin-Chatelin, F., Frayssé, V.: Lectures on Finite Precision Computations. Software, Environments and Tools. SIAM, Philadelphia (1996)CrossRefGoogle Scholar
  9. 9.
    Chandrasekaran, S., Gu, M., Xia, J., Zhu, J.: A fast QR algorithm for companion matrices. In: Recent Advances in Matrix and Operator Theory, pp. 111–143. Basel, Birkhäuser (2008)Google Scholar
  10. 10.
    Chen, T.C.: Improving Laguerre’s method to cope with symmetry pitfalls in polynomial root-finding. In: Proceedings of the Eigth International Colloquium on Differential Equations, pp. 105–110. VSP, Utrecht (1998)Google Scholar
  11. 11.
    Ehrlich, L.W.: A modified Newton method for polynomials. Commun. ACM 10(2), 107–108 (1967)CrossRefGoogle Scholar
  12. 12.
    Foster, L.V.: Generalizations of Laguerre’s method: higher order methods. SIAM J. Numer. Anal. 18(6), 1004–1018 (1981)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Goedecker, S.: Remarks on algorithms to find roots of polynomials. SIAM J. Sci. Comput. 15(5), 1059–1063 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. 23, 5–48 (1991)CrossRefGoogle Scholar
  15. 15.
    Grad, J., Zakrajšek, E.: LR algorithm with Laguerre shift for symmetric tridiagonal matrices. Comput. J. 15(3), 268–270 (1972)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1977)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Hansen, E., Patrick, M., Rusnak, J.: Some modifications of Laguerre’s method. BIT 17(4), 409–417 (1977)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (2002)CrossRefGoogle Scholar
  19. 19.
    Jenkins, M.A., Traub, J.F.: Principles for testing polynomial zerofinding programs. ACM Trans. Math. Softw. 1(1), 26–34 (1975)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Kahan, W.: Laguerre’s method and a circle which contains at least one zero of a polynomial. SIAM J. Numer. Anal, 4(3) (1967)Google Scholar
  21. 21.
    Laguerre, E.N.: Oeuvres de Laguerre. Gauthier-Villars and Fils, Paris (1898)zbMATHGoogle Scholar
  22. 22.
    Lancaster, P.: Lambda-Matrices and Vibrating Systems International Series of Monographs on Pure and Applied Mathematics, vol. 94. Pergamon, Oxford (1966)Google Scholar
  23. 23.
    Lenard, C.T.: Laguerre’s iteration and the method of traces for eigenproblems. Appl. Math. Lett. 3(3), 73–74 (1990)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Leuze, M.R.: A hybrid Laguerre method. BIT 23(1), 132–138 (1983)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Maehly, V.H.: Zur iterativen auflösung algebraisher gleichungen. Z. Angew. Math. Phys. 5, 260–263 (1954)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Mekwi, W.: Iterative Methods for Roots of Polynomials. Master’s thesis, University of Oxford, Oxford (2001)Google Scholar
  27. 27.
    MPFUN: Multiprecision Software, http://www.netlib.org/mpfun. Accessed 18 June 2018 (2005)
  28. 28.
    NAG: Nag Library, Mark 26, https://www.nag.com/numeric/fl/nagdoc_latest/html/c02/c02aff.html. Accessed 27 May 2018 (2017)
  29. 29.
    Pan, V.Y.: Solving a polynomial equation: some history and recent progress. SIAM Rev. 39(2), 187–220 (1997)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Parlett, B.: Laguerre’s method applied to the matrix eigenvalue problem. Math. Comput. 18(87), 464–485 (1964)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Pellet, A.E.: Sur un mode de séparation des racines des équations et la formule de lagrange. Bull. Sci. Math. 5(2), 393–395 (1881)zbMATHGoogle Scholar
  32. 32.
    Petković, M.S., Ilić, S., Rančić, L.: The convergence of a family of parallel zero-finding methods. Comput. Math. Appl. 48, 455–467 (2004)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Petković, M.S., Ilić, S., Tričković, S.: A family of simultaneous zero finding methods. Comput. Math. Appl. 34(10), 49–59 (1997)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Petković, M.S., Petković, L., živković, D.: Laguerre-like methods for the simultaneous approximation of polynomial zeros. In: Topics in Numerical Analysis, pp. 189–209. Springer, Vienna (2001)Google Scholar
  35. 35.
    Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical Recipes: The Art of Scientific Computing, 3rd edn. Cambridge University Press, New York (2007)zbMATHGoogle Scholar
  36. 36.
    Redish, K.A.: On Laguerre’s method. Int. J. Math. Educ. Sci. Tech. 5(1), 91–102 (1974)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Sitton, G.A., Burrus, C.S., Fox, J.W., Treitel, S.: Factoring very-high-degree polynomials. IEEE Sig. Proc. Mag. 20, 27–42 (2003)CrossRefGoogle Scholar
  38. 38.
    Smith, B.T.: A Zero Finding Algorithm using Laguerre’s Method. Master’s thesis, University of Toronto, Toronto (1967)Google Scholar
  39. 39.
    Walsh, J.L.: On Pellet’s theorem concerning the roots of a polynomial. Ann. Math. 26, 59–64 (1924)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Prentice-Hall Inc (1963)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Mathematics and Computer Science DepartmentDavidson CollegeDavidsonUSA

Personalised recommendations

Cite article

Buy options