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New Practical Advances in Polynomial Root Clustering

  • Rémi ImbachEmail author
  • Victor Y. Pan
Conference paper
  • 27 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11989)

Abstract

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial p of degree d with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for multiple roots of a polynomial given by a black box for the approximation of its coefficients, and their complexity decreases at least proportionally to the number of roots in a region of interest (ROI) on the complex plane, such as a disc or a square, but we greatly strengthen the main ingredient of the previous algorithms. We build the foundation for a new counting test that essentially amounts to the evaluation of a polynomial p and its derivative \(p'\), which is a major benefit, e.g., for sparse polynomials p. Moreover with evaluation at about \(\log (d)\) points (versus the previous record of order d) we output correct number of roots in a disc whose contour has no roots of p nearby. Our second and less significant contribution concerns subdivision algorithms for polynomials with real coefficients. Our tests demonstrate the power of the proposed algorithms.

References

  1. [BF00]
    Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms 23(2), 127–173 (2000)MathSciNetCrossRefGoogle Scholar
  2. [BR14]
    Bini, D.A., Robol, L.: Solving secular and polynomial equations: a multiprecision algorithm. J. Comput. Appl. Math. 272, 276–292 (2014)MathSciNetCrossRefGoogle Scholar
  3. [BSS+16]
    Becker, R., Sagraloff, M., Sharma, V., Xu, J., Yap, C.: Complexity analysis of root clustering for a complex polynomial. In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2016, pp. 71–78. ACM, New York (2016)Google Scholar
  4. [BSSY18]
    Becker, R., Sagraloff, M., Sharma, V., Yap, C.: A near-optimal subdivision algorithm for complex root isolation based on Pellet test and Newton iteration. J. Symb. Comput. 86, 51–96 (2018)MathSciNetCrossRefGoogle Scholar
  5. [HG69]
    Henrici, P., Gargantini, I.: Uniformly convergent algorithms for the simultaneous approximation of all zeros of a polynomial. In: Constructive Aspects of the Fundamental Theorem of Algebra, pp. 77–113. Wiley-Interscience, New York (1969)Google Scholar
  6. [IPY18]
    Imbach, R., Pan, V.Y., Yap, C.: Implementation of a near-optimal complex root clustering algorithm. Math. Soft. - ICMS 2018, 235–244 (2018)zbMATHGoogle Scholar
  7. [KRS16]
    Kobel, A., Rouillier, F., Sagraloff, M.: Computing real roots of real polynomials ... and now for real! In: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, ISSAC 2016, pp. 303–310. ACM, New York (2016)Google Scholar
  8. [Pan00]
    Pan, V.Y.: Approximating complex polynomial zeros: modified Weyl’s quadtree construction and improved newton’s iteration. J. Complex. 16(1), 213–264 (2000)MathSciNetCrossRefGoogle Scholar
  9. [Pan02]
    Pan, V.Y.: Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding. J. Symb. Comput. 33(5), 701–733 (2002)MathSciNetCrossRefGoogle Scholar
  10. [Pan18]
    Pan, V.Y.: Old and new nearly optimal polynomial root-finders. arXiv preprint arXiv:1805.12042 (2018)
  11. [PT13]
    Pan, V.Y., Tsigaridas, E.P.: On the Boolean complexity of real root refinement. In: Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ISSAC 2013, pp. 299–306. ACM, New York (2013)Google Scholar
  12. [PT16]
    Pan, V.Y., Tsigaridas, E.P.: Nearly optimal refinement of real roots of a univariate polynomial. J. Symb. Comput 74, 181–204 (2016)MathSciNetCrossRefGoogle Scholar
  13. [Ren87]
    Renegar, J.: On the worst-case arithmetic complexity of approximating zeros of polynomials. J. Complex. 3(2), 90–113 (1987)MathSciNetCrossRefGoogle Scholar
  14. [Sch82]
    Schönhage, A.: The fundamental theorem of algebra in terms of computational complexity. Manuscript. University of Tübingen, Germany (1982)Google Scholar
  15. [SM16]
    Sagraloff, M., Mehlhorn, K.: Computing real roots of real polynomials. J. Symb. Comput. 73, 46–86 (2016)MathSciNetCrossRefGoogle Scholar
  16. [XY19]
    Xu, J., Yap, C.: Effective subdivision algorithm for isolating zeros of real systems of equations, with complexity analysis. arXiv preprint (2019). arXiv:1905.03505
  17. [ZZ19]
    Zaderman, V., Zhao, L.: Counting roots of a polynomial in a convex compact region by means of winding number calculation via sampling. arXiv preprint arXiv:1906.10805 (2019)

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.Lehman College and the Graduate CenterCity University of New YorkNew YorkUSA

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