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


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.

This is a preview of subscription content, log in to check access.


  1. 1.

    Aberth, O.: Iteration methods for finding all zeros of a polynomial simultaneously. Math. Comput. 27(122), 339–344 (1973)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Andrew, A.M.: Another efficient algorithm for convex hulls in two dimensions. Info. Proc. Lett. 9(15), 216–219 (1979)

    Article  Google 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)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Bini, D.A.: Numerical computation of polynomial zeros by means of Aberth’s method. Numer. Algor. 13, 179–200 (1996)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial root finder. Numer. Algor. 23, 127–173 (2000)

    Article  Google Scholar 

  6. 6.

    Bini, D.A., Robol, L.: Solving secular and polynomial equations: a multiprecision algorithm. J. Comput. Appl. Math. 272, 276–292 (2014)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Börsch-Supan, W.: A posteriori error bounds for the zeros of polynomials. Numer. Math. 5, 380–398 (1963)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Chaitin-Chatelin, F., Frayssé, V.: Lectures on Finite Precision Computations. Software, Environments and Tools. SIAM, Philadelphia (1996)

    Google 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)

  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)

  11. 11.

    Ehrlich, L.W.: A modified Newton method for polynomials. Commun. ACM 10(2), 107–108 (1967)

    Article  Google Scholar 

  12. 12.

    Foster, L.V.: Generalizations of Laguerre’s method: higher order methods. SIAM J. Numer. Anal. 18(6), 1004–1018 (1981)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Goedecker, S.: Remarks on algorithms to find roots of polynomials. SIAM J. Sci. Comput. 15(5), 1059–1063 (1994)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Goldberg, D.: What every computer scientist should know about floating-point arithmetic. ACM Comput. Surv. 23, 5–48 (1991)

    Article  Google Scholar 

  15. 15.

    Grad, J., Zakrajšek, E.: LR algorithm with Laguerre shift for symmetric tridiagonal matrices. Comput. J. 15(3), 268–270 (1972)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1977)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Hansen, E., Patrick, M., Rusnak, J.: Some modifications of Laguerre’s method. BIT 17(4), 409–417 (1977)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM, Philadelphia (2002)

    Google Scholar 

  19. 19.

    Jenkins, M.A., Traub, J.F.: Principles for testing polynomial zerofinding programs. ACM Trans. Math. Softw. 1(1), 26–34 (1975)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Laguerre, E.N.: Oeuvres de Laguerre. Gauthier-Villars and Fils, Paris (1898)

    Google 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)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Leuze, M.R.: A hybrid Laguerre method. BIT 23(1), 132–138 (1983)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Maehly, V.H.: Zur iterativen auflösung algebraisher gleichungen. Z. Angew. Math. Phys. 5, 260–263 (1954)

    MathSciNet  Article  Google 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, Accessed 18 June 2018 (2005)

  28. 28.

    NAG: Nag Library, Mark 26, 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)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Parlett, B.: Laguerre’s method applied to the matrix eigenvalue problem. Math. Comput. 18(87), 464–485 (1964)

    MathSciNet  MATH  Google 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)

    MATH  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Google Scholar 

  36. 36.

    Redish, K.A.: On Laguerre’s method. Int. J. Math. Educ. Sci. Tech. 5(1), 91–102 (1974)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  38. 38.

    Smith, B.T.: A Zero Finding Algorithm using Laguerre’s Method. Master’s thesis, University of Toronto, Toronto (1967)

  39. 39.

    Walsh, J.L.: On Pellet’s theorem concerning the roots of a polynomial. Ann. Math. 26, 59–64 (1924)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Wilkinson, J.H.: Rounding Errors in Algebraic Processes. Prentice-Hall Inc (1963)

Download references


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.

Author information



Corresponding author

Correspondence to Thomas R. Cameron.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cameron, T.R. An effective implementation of a modified Laguerre method for the roots of a polynomial. Numer Algor 82, 1065–1084 (2019).

Download citation


  • Laguerre’s method
  • Polynomial roots
  • Mathematical software

Mathematics Subject Classification (2010)

  • 26C10
  • 65H04
  • 65Y20