Skip to main content
SpringerLink
Log in

Old and New Nearly Optimal Polynomial Root-Finders

  • Conference paper
  • First Online:
Book cover Computer Algebra in Scientific Computing (CASC 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11661))

Included in the following conference series:

Abstract

Univariate polynomial root-finding has been studied for four millennia and still remains the subject of intensive research. Hundreds if not thousands of efficient algorithms for this task have been proposed and analyzed. Two nearly optimal solution algorithms have been devised in 1995 and 2016, based on recursive factorization of a polynomial and subdivision iterations, respectively, but both of them are superseded in practice by Ehrlich’s functional iterations. By combining factorization techniques with Ehrlich’s and subdivision iterations we devise a variety of new root-finders. They match or supersede the known algorithms in terms of their estimated complexity for root-finding on the complex plane, in a disc, and in a line segment and promise to be practically competitive.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Required precision and Boolean time are smaller by a factor of d, the degree of the input polynomial, at the stage of numerical polynomial factorization, which has various important applications to modern computations, besides root-finding, e.g., to time series analysis, Wiener filtering, noise variance estimation, co-variance matrix computation, and the study of multi-channel systems (see Wilson [59], Box and Jenkins [7], Barnett [3], Demeure and Mullis [16] and [17], Van Dooren [56]).

  2. 2.

    Some competition came in 2001 from the package EigenSolve of [20], but the latest version of MPSolve of [10] has combined the benefits of both packages.

  3. 3.

    We count m times a root of multiplicity m.

  4. 4.

    Here and hereafter we write \(\tilde{O}(s)\) for O(s) defined up to a poly-logarithmic factor in s.

  5. 5.

    The paper [10] elaborates upon expression of Ehrlich’s iterations via secular equation, shows significant numerical benefits of root-finding by using this expression, and traces the previous study of this approach back to [6].

  6. 6.

    Furthermore we may have ITER\(_f<\)ITER\(_p\) because of the decrease of the maximal distance between a pair of roots and of the number and sizes of root clusters in the transition from p to the polynomial f(x) of a smaller degree w.

  7. 7.

    [50] supplies estimates for the working precision in such a recursive process, which ensure the bound \(1/2^b\) on the errors of the output approximations to the roots of p.

  8. 8.

    The wild roots are much less numerous than the tame roots in a typical partition of a root set observed in Ehrlich’s, Weierstrass’s and other functional iterations that simultaneously approximate all roots of p as well as in Newton’s iteration in [54]. Consequently the coefficient growth and the loss of sparseness are not dramatic in the transition to the factors defined by the wild roots.

  9. 9.

    The algorithms of [11] and [12] are quite similar to one another.

  10. 10.

    The DLS algorithms (cf. [42, Appendices A and B]) approximates a factor f at a nearly optimal Boolean cost if the disc \(D'_i\) is \(\theta \)-isolated for isolation ratio \(\theta =1+g/\log ^{h}(d)\), a positive constant g and a real constant h. Such an isolation ratio is smaller than those required in [11, 35, 49] and [12] and thus can be ensured by means of performing fewer subdivision steps.

  11. 11.

    Then again with such a delay we bound the overall cost of all deflation steps (cf. Example 1), avoid coefficient growth and do not lose sparseness.

References

  1. Bell, E.T.: The Development of Mathematics. McGraw-Hill, New York (1940)

    MATH  Google Scholar 

  2. Boyer, C.A.: A History of Mathematics. Wiley, New York (1968)

    MATH  Google Scholar 

  3. Barnett, S.: Polynomial and Linear Control Systems. Marcel Dekker, New York (1983)

    MATH  Google Scholar 

  4. Bini, D.A.: Parallel solution of certain Toeplitz linear systems. SIAM J. Comput. 13(2), 268–279 (1984)

    Article  MathSciNet  Google Scholar 

  5. Bini, D.A., Fiorentino, G.: Design, Analysis, and Implementation of a Multiprecision Polynomial Rootfinder. Numer. Algorithms 23, 127–173 (2000)

    Article  MathSciNet  Google Scholar 

  6. Bini, D.A., Gemignani, L., Pan, V.Y.: Inverse power and Durand/Kerner iteration for univariate polynomial root-finding. Comput. Math. Appl. 47(2/3), 447–459 (2004)

    Article  MathSciNet  Google Scholar 

  7. Box, G.E.P., Jenkins, G.M.: Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco (1976)

    MATH  Google Scholar 

  8. Bini, D., Pan, V.Y.: Computing matrix eigenvalues and polynomial zeros where the output is real. SIAM J. Comput. 27(4), 1099–1115 (1998). Proc. version. In: SODA 1991, pp. 384–393. ACM Press, NY, and SIAM Publ., Philadelphia (1991)

    Google Scholar 

  9. Bini, D., Pan, V.Y.: Graeffe’s, Chebyshev, and Cardinal’s processes for splitting a polynomial into factors. J. Complex. 12, 492–511 (1996)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Becker, R., Sagraloff, M., Sharma, V., Xu, J., Yap, C.: Complexity analysis of root clustering for a complex polynomial. In: International Symposium on Symbolic and Algebraic Computation (ISSAC 2016), pp. 71–78. ACM Press, New York (2016)

    Google Scholar 

  12. Becker, R., Sagraloff, M., Sharma, V., Yap, C.: A near-optimal subdivision algorithm for complex root isolation based on the Pellet test and Newton iteration. J. Symb. Comput. 86, 51–96 (2018)

    Article  MathSciNet  Google Scholar 

  13. Ben-Or, M., Tiwari, P.: Simple algorithms for approximating all roots of a polynomial with real roots. J. Complex. 6(4), 417–442 (1990)

    Article  MathSciNet  Google Scholar 

  14. Du, Q., Jin, M., Li, T.Y., Zeng, Z.: The quasi-Laguerre iteration. Math. Comput. 66(217), 345–361 (1997)

    Article  MathSciNet  Google Scholar 

  15. Delves, L.M., Lyness, J.N.: A numerical method for locating the zeros of an analytic function. Math. Comput. 21, 543–560 (1967)

    Article  MathSciNet  Google Scholar 

  16. Demeure, C.J., Mullis, C.T.: The Euclid algorithm and the fast computation of cross-covariance and autocovariance sequences. IEEE Trans. Acoust. Speech Signal Process. 37, 545–552 (1989)

    Article  Google Scholar 

  17. Demeure, C.J., Mullis, C.T.: A Newton-Raphson method for moving-average spectral factorization using the Euclid algorithm. IEEE Trans. Acoust. Speech Signal Process. 38, 1697–1709 (1990)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  19. Emiris, I.Z., Pan, V.Y., Tsigaridas, E.: Algebraic algorithms. In: Tucker, A.B., Gonzales, T., Diaz-Herrera, J.L. (eds.) Computing Handbook. Computer Science and Software Engineering, 3rd edn., vol. I, Chap. 10, pp. 10-1–10-40. Taylor and Francis Group (2014)

    Google Scholar 

  20. Fortune, S.: J. Symbol. Comput. 33(5), 627–646 (2002). Proc. version in Proc. Intern. Symp. on Symbolic and Algebraic Computation An Iterated Eigenvalue Algorithm for Approximating Roots of Univariate Polynomials, (ISSAC 2001), 121–128, ACM Press, New York (2001)

    Google Scholar 

  21. Householder, A.S.: Dandelin, Lobachevskii, or Graeffe? Amer. Math. Mon. 66, 464–466 (1959)

    MATH  Google Scholar 

  22. Henrici, P.: Applied and computational complex analysis. In: Power Series, Integration, Conformal Mapping, Location of Zeros, vol. 1. Wiley, New York (1974)

    Google Scholar 

  23. Henrici, P., Gargantini, I.: Uniformly convergent algorithms for the simultaneous approximation of all zeros of a polynomial. In: Dejon, B., Henrici, P. (eds.) Constructive Aspects of the Fundamental Theorem of Algebra. Wiley, New York (1969)

    MATH  Google Scholar 

  24. Imbach, R., Pan, V.Y., Yap, C.: Implementation of a near-optimal complex root clustering algorithm. In: Davenport, J.H., Kauers, M., Labahn, G., Urban, J. (eds.) ICMS 2018. LNCS, vol. 10931, pp. 235–244. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96418-8_28

    Chapter  Google Scholar 

  25. Inbach, R., Pan, V.Y., Yap, C., Kotsireas, I.S., Zaderman, V.: Root-finding with implicit deflation. In: Proceedings CASC 2019. arxiv:1606.01396. Accepted 21 May 2019

  26. Kirrinnis, P.: Polynomial factorization and partial fraction decomposition by simultaneous Newton’s iteration. J. Complex. 14, 378–444 (1998)

    Article  MathSciNet  Google Scholar 

  27. Kobel, A., Rouillier, F., Sagraloff, M.: Computing real roots of real polynomials ... and now for real! In: The International Symposium on Symbolic and Algebraic Computation (ISSAC 2016), pp. 301–310. ACM Press, New York (2016)

    Google Scholar 

  28. McNamee, J.M.: Numerical Methods for Roots of Polynomials, Part I, p. XIX+354. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  29. Moenck, R., Borodin, A.: Fast modular transforms via division., In: Proceedings of 13th Annual Symposium on Switching and Automata Theory (SWAT 1972), pp. 90–96. IEEE Computer Society Press (1972)

    Google Scholar 

  30. McNamee, J.M., Pan, V.Y.: Numerical Methods for Roots of Polynomials, Part II, p. XXI+728. Elsevier, Amsterdam (2013)

    MATH  Google Scholar 

  31. Neff, C.A., Reif, J.H.: An \(o(n^{1 +\epsilon })\) algorithm for the complex root problem. In: Proceedings 35th Annual Symposium on Foundations of Computer Science (FOCS 1994), pp. 540–547. IEEE Computer Society Press (1994)

    Google Scholar 

  32. Pan, V.Y.: Optimal (up to polylog factors) sequential and parallel algorithms for approximating complex polynomial zeros. In: Proceedings of 27th Annual ACM Symposium on Theory of Computing (STOC 1995), pp. 741–750. ACM Press, New York (1995)

    Google Scholar 

  33. Pan, V.Y.: Solving a polynomial equation: some history and recent progress. SIAM Rev. 39(2), 187–220 (1997)

    Article  MathSciNet  Google Scholar 

  34. Pan, V.Y.: Solving polynomials with computers. Am. Sci. 86, 62–69 (1998)

    Article  Google Scholar 

  35. Pan, V.Y.: Approximation of complex polynomial zeros: modified quadtree (Weyl’s) construction and improved Newton’s iteration. J. Complex. 16(1), 213–264 (2000)

    Article  Google Scholar 

  36. Pan, V.Y.: Structured Matrices and Polynomials: Unified Superfast Algorithms. Birkhäuser/Springer, Boston/New York (2001). https://doi.org/10.1007/978-1-4612-0129-8

    Book  MATH  Google Scholar 

  37. Pan, V.Y.: Univariate polynomials: nearly optimal algorithms for factorization and rootfinding. J. Symb. Comput. 33(5), 701–733 (2002)

    Article  Google Scholar 

  38. Pan, V.Y.: Transformations of matrix structures work again. Linear Algebra Appl. 465, 1–32 (2015)

    Article  MathSciNet  Google Scholar 

  39. Pan, V.Y.: Root-finding with Implicit Deflation. arXiv:1606.01396, Accepted 4 June 2016

  40. Pan, V.Y.: Simple and nearly optimal polynomial root-finding by means of root radii approximation. In: Kotsireas, I., Martinez-Moro, E. (eds.) ACA 2015. SPMS, vol. 198, pp. 329–340. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-56932-1_23

    Chapter  Google Scholar 

  41. Pan, V.Y.: Fast approximate computations with Cauchy matrices and polynomials. Math. Comput. 86, 2799–2826 (2017)

    Article  MathSciNet  Google Scholar 

  42. Pan, V.Y.: Old and new nearly optimal polynomial root-finders, In: Proceedings of CASC (2019). Also arxiv: 1805.12042 May 2019

  43. Pan, V.Y., Sadikou, A., Landowne, E.: Univariate polynomial division with a remainder by means of evaluation and interpolation. In: Proceedings of 3rd IEEE Symposium on Parallel and Distributed Processing, pp. 212–217. IEEE Computer Society Press, Los Alamitos (1991)

    Google Scholar 

  44. Pan, V.Y., Sadikou, A., Landowne, E.: Polynomial division with a remainder by means of evaluation and interpolation. Inform. Process. Lett. 44, 149–153 (1992)

    Article  MathSciNet  Google Scholar 

  45. Pan, V.Y., Tsigaridas, E.P.: Nearly optimal refinement of real roots of a univariate polynomial. J. Symb. Comput. 74, 181–204 (2016). Proceedings version. In: Kauers, M. (ed.) Proc. ISSAC 2013, pp. 299–306. ACM Press, New York (2013)

    Google Scholar 

  46. Pan, V.Y., Tsigaridas, E.P.: Nearly optimal computations with structured matrices. Theor. Comput. Sci. 681, 117–137 (2017)

    Article  MathSciNet  Google Scholar 

  47. Pan, V.Y., Zhao, L.: Real root isolation by means of root radii approximation. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2015. LNCS, vol. 9301, pp. 349–360. Springer, Heidelberg (2015). https://doi.org/10.1007/978-3-319-24021-3_26

    Chapter  Google Scholar 

  48. Pan, V.Y., Zhao, L.: Real polynomial root-finding by means of matrix and polynomial iterations. Theor. Comput. Sci. 681, 101–116 (2017)

    Article  MathSciNet  Google Scholar 

  49. Renegar, J.: On the worst-case arithmetic complexity of approximating zeros of polynomials. J. Complex. 3(2), 90–113 (1987)

    Article  MathSciNet  Google Scholar 

  50. Schönhage, A.: The Fundamental Theorem of Algebra in Terms of Computational Complexity. Department of Mathematics. University of Tübingen, Tübingen, Germany (1982)

    Google Scholar 

  51. Schönhage, A.: Asymptotically fast algorithms for the numerical muitiplication and division of polynomials with complex coefficients. In: Calmet, J. (ed.) EUROCAM 1982. LNCS, vol. 144, pp. 3–15. Springer, Heidelberg (1982). https://doi.org/10.1007/3-540-11607-9_1

    Chapter  Google Scholar 

  52. Schönhage, A.: Quasi GCD computations. J. Complex. 1, 118–137 (1985)

    Article  MathSciNet  Google Scholar 

  53. Schleicher, D.: Private communication (2018)

    Google Scholar 

  54. Schleicher, D., Stoll, R.: Newton’s method in practice: finding all roots of polynomials of degree one million efficiently. Theor. Comput. Sci. 681, 146–166 (2017)

    Article  MathSciNet  Google Scholar 

  55. Tilli, P.: Convergence conditions of some methods for the simultaneous computation of polynomial zeros. Calcolo 35, 3–15 (1998)

    Article  MathSciNet  Google Scholar 

  56. Van Dooren, P.: Some numerical challenges in control theory. In: Van Dooren, P., Wyman, B. (eds.) Linear Algebra for Control Theory. The IMA Volumes in Mathematics and its Applications, vol. 62. Springer, New York (1994). https://doi.org/10.1007/978-1-4613-8419-9_12

    Chapter  Google Scholar 

  57. Weierstrass, K.: Neuer Beweis des Fundamentalsatzes der Algebra. Mathematische Werke, Bd, vol. III, pp. 251–269. Mayer und Mueller, Berlin (1903)

    Google Scholar 

  58. Weyl, H.: Randbemerkungen zu Hauptproblemen der Mathematik. II. Fundamentalsatz der Algebra und Grundlagen der Mathematik. Math. Z. 20, 131–151 (1924)

    Google Scholar 

  59. Wilson, G.T.: Factorization of the covariance generating function of a pure moving-average. SIAM J. Numer. Anal. 6, 1–7 (1969)

    Article  MathSciNet  Google Scholar 

  60. Werner, W.: Some improvements of classical iterative methods for the solution of nonlinear equations. In: Allgower, E.L., Glashoff, K., Peitgen, H.-O. (eds.) Numerical Solution of Nonlinear Equations. LNM, vol. 878, pp. 426–440. Springer, Heidelberg (1981). https://doi.org/10.1007/BFb0090691

    Chapter  Google Scholar 

Download references

Acknowledgements

This research has been supported by NSF Grants CCF–1563942 and CCF–1733834 and PSC CUNY Award 69813 00 48.

Author information

Authors and Affiliations

  1. Department of Computer Science, Lehman College of the City University of New York, Bronx, NY, 10468, USA

    Victor Y. Pan

  2. Ph.D. Programs in Mathematics and Computer Science, The Graduate Center of the City University of New York, NewYork, NY, 10036, USA

    Victor Y. Pan

Authors
  1. Victor Y. Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Victor Y. Pan .

Editor information

Editors and Affiliations

  1. Coventry University, Coventry, UK

    Matthew England

  2. University of Kassel, Kassel, Germany

    Wolfram Koepf

  3. Plekhanov Russian University of Economics, Moscow, Russia

    Timur M. Sadykov

  4. University of Kassel, Kassel, Germany

    Werner M. Seiler

  5. Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Pan, V.Y. (2019). Old and New Nearly Optimal Polynomial Root-Finders. In: England, M., Koepf, W., Sadykov, T., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2019. Lecture Notes in Computer Science(), vol 11661. Springer, Cham. https://doi.org/10.1007/978-3-030-26831-2_26

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-26831-2_26

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26830-5

  • Online ISBN: 978-3-030-26831-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation