Root-Refining for a Polynomial Equation

  • Victor Y. Pan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7442)


Polynomial root-finding usually consists of two stages. At first a crude approximation to a root is slowly computed; then it is much faster refined by means of the same or distinct iterations. The efficiency of computing an initial approximation resists formal study, and the users employ empirical data. In contrast, the efficiency of refinement is formally measured by the classical concept q 1/α where q is the convergence order and α is the number of function evaluations per iteration. To cover iterations not reduced to function evaluations alone, e.g., ones simultaneously refining n approximations to all n roots of a degree n polynomial, we let d denote the number of arithmetic operations involved in an iteration divided by 2n because we can evaluate such a polynomial at a point by using 2n operations. For this task we employ recursive polynomial factorization to yield refinement with the efficiency \(2^{cn/\log^2 n}\) for a positive constant c. For large n this is a dramatic increase versus the record efficiency 2 of refining an approximation to a single root of a polynomial. The advance could motivate practical use of the proposed root-refiners.


Root-refining Efficiency Polynomial factorization 


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  1. 1.
    Aberth, O.: Iteration Methods For Finding All Zeros of a Polynomial Simultaneously. Mathematics of Computation 27(122), 339–344 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Aitken, A.C.: On Bernoulli’s numerical solution of algebraic equations. Proc. Roy. Soc. Edin. 46, 289–305 (1926)zbMATHGoogle Scholar
  3. 3.
    Barnett, S.: Polynomial and Linear Control Systems. Marcel Dekker, NY (1983)Google Scholar
  4. 4.
    Bell, E.T.: The Development of Mathematics. McGraw-Hill, New York (1940)Google Scholar
  5. 5.
    Bini, D., Pan, V.Y.: Polynomial and Matrix Computations, Fundamental Algorithms, vol. 1. Birkhäuser, Boston (1994)zbMATHCrossRefGoogle Scholar
  6. 6.
    Bollobàs, B., Lackmann, M., Schleicher, D.: A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton’s method. Math. of Computation (in press, 2012), arXiv:1009.1843Google Scholar
  7. 7.
    Box, G.E.P., Jenkins, G.M.: Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco (1976)Google Scholar
  8. 8.
    Boyer, C.A.: A History of Mathematics. Wiley, New York (1968)zbMATHGoogle Scholar
  9. 9.
    Curry, J.H.: On zero finding methods of higher order from data at one point. J. of Complexity 5, 219–237 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Durand, E.: Equations du type F(x) = 0: Racines d’un polynome, In Solutions numérique équation algébrique, Masson, Paris, vol. 1 (1960)Google Scholar
  11. 11.
    Demeure, C.J., Mullis, C.T.: A Newton–Raphson method for moving-average spectral factorization using the Euclid algorithm. IEEE Trans. Acoust., Speech, Signal Processing 38, 1697–1709 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Ehrlich, L.W.: A modified Newton method for polynomials. Comm. of ACM 10, 107–108 (1967)zbMATHCrossRefGoogle Scholar
  13. 13.
    Faugère, J.C.: A new efficient algorithm for computing Gröbner bases without reduction to zero (F5). In: Proc. ISSAC 2002, pp. 75–83. ACM Press, NY (2002)Google Scholar
  14. 14.
    Householder, A.S.: Dandelin, Lobachevskii, or Graeffe. American Mathematical Monthly 66, 464–466 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kantorovich, L.V., Akilov, G.P.: Functional Analysis. Pergamon Press (1982)Google Scholar
  16. 16.
    Kerner, I.O.: Ein Gesamtschrittverfahren zur Berechnung der Nullstellen von Polynomen. Numerische Mathematik 8, 290–294 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Kim, M.-H.: Computational complexity of the Euler type algorithms for the roots of complex polynomials. PhD Thesis, City University of New York (1985)Google Scholar
  18. 18.
    Kirrinnis, P.: Polynomial factorization and partial fraction decomposition by simultaneous Newton’s iteration. J. of Complexity 14, 378–444 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Mahler, K.: An Inequality for the Discriminant of a Polynomial. Michigan Math. Journal 11, 257–262 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    McNamee, J.M.: A 2002 update of the supplementary bibliography on root of polynomials. J. Comput. Appl. Math. 142, 433–434 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    McNamee, J.M.: Numerical Methods for Roots of Polynomials (Part 1). Elsevier, Amsterdam (2007)Google Scholar
  22. 22.
    McNamee, J.M., Pan, V.Y.: Efficient polynomial root-refiners: survey and new record estimates. Computers and Math. with Applics. 63, 239–254 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Mourrain, B., Pan, V.Y.: Multivariate polynomials, duality and structured matrices. J. of Complexity 16(1), 110–180 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Muller, D.E.: A method for solving algebraic equations using an automatic computer. Math. Tables Aids Comput. 10, 208–215 (1956)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Neff, C.A., Reif, J.H.: An o(n 1 + ε) algorithm for the complex root problem. In: Proc. STOC 1994, pp. 540–547. IEEE Computer Society Press (1994)Google Scholar
  26. 26.
    Ostrowski, A.M.: Recherches sur la méthode de Graeffe et les zéros des polynomes et des sèries de Laurent. Acta Math 72, 99–257 (1940)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Ostrowski, A.M.: Solution of Equations and Systems of Equations, 2nd edn. Academic Press, New York (1966)zbMATHGoogle Scholar
  28. 28.
    Pan, V.Y.: Optimal (up to polylog factors) sequential and parallel algorithms for approximating complex polynomial zeros. In: Proc. 27th Ann. ACM Symp. on Theory of Computing, pp. 741–750. ACM Press, New York (1995)Google Scholar
  29. 29.
    Pan, V.Y.: Optimal and nearly optimal algorithms for approximating polynomial zeros. Computers and Math. with Applications 31(12), 97–138 (1996)zbMATHCrossRefGoogle Scholar
  30. 30.
    Pan, V.Y.: Solving a polynomial equation: some history and recent progress. SIAM Review 39(2), 187–220 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Pan, V.Y.: Solving polynomials with computers. American Scientist 86 ( January-February 1998)Google Scholar
  32. 32.
    Pan, V.Y.: Univariate polynomials: nearly optimal algorithms for factorization and rootfinding. J. Symbolic Computation 33(5), 701–733 (2002)zbMATHCrossRefGoogle Scholar
  33. 33.
    Pan, V.Y.: Amended DSeSC Power Method for polynomial root-finding. Computers and Math (with Applications) 49(9-10), 1515–1524 (2005)zbMATHCrossRefGoogle Scholar
  34. 34.
    Pan, V.Y., Zheng, A.–L.: New progress in real and complex polynomial root-finding. Computers and Mathematics with Applications 61, 1305–1334 (2011a)MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Pan, V.Y., Zheng, A.–L.: Root-finding by expansion with independent constraints. Computers and Mathematics with Applications 62, 3164–3182 (2011b)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Petkovic, M.S., Herceg, D.: Point estimation of simultaneous methods for solving polynomial equations: a survey. Computers Math. with Applics. 136, 183–207 (2001)MathSciNetGoogle Scholar
  37. 37.
    Renegar, J.: On the worst-case arithmetic complexity of approximating zeros of polynomials. J. of Complexity 3, 90–113 (1987)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    Schönhage, A.: The fundamental theorem of algebra in terms of computational complexity. Department of Math., University of Tübingen, Germany (1982)Google Scholar
  39. 39.
    Smale, S.: The fundamental theorem of algebra and complexity theory. Bulletin of the American Mathematical Society 4, 1–36 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Smale, S.: Newton’s method estimates from data at one point. In: Ewing, R.E., Cross, K.I., Martin, C.F. (eds.) The Merging Disciplines: New Directions in Pure, Applied and Computational Math., pp. 185–196. Springer (1986)Google Scholar
  41. 41.
    Van Dooren, P.M.: Some numerical challenges in control theory. Linear Algebra for Control Theory IMA Vol. Math. Appl (1994)Google Scholar
  42. 42.
    Weierstrass, K.: Neuer Beweis des Fundamentalsatzes der Algebra. Mathematische Werke, Band III, Mayer und Müller, Berlin, 251–269 (1903)Google Scholar
  43. 43.
    Wilson, G.T.: Factorization of the covariance generating function of a pure moving-average process. SIAM J. on Numerical Analysis 6, 1–7 (1969)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Victor Y. Pan
    • 1
  1. 1.Department of Mathematics and Computer ScienceLehman College and the Graduate Center of the City University of New YorkBronxUSA

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