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Implementation of a Near-Optimal Complex Root Clustering Algorithm

  • Rémi ImbachEmail author
  • Victor Y. Pan
  • Chee Yap
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10931)

Abstract

We describe Ccluster, a software for computing natural \(\varepsilon \)-clusters of complex roots in a given box of the complex plane. This algorithm from Becker et al. (2016) is near-optimal when applied to the benchmark problem of isolating all complex roots of an integer polynomial. It is one of the first implementations of a near-optimal algorithm for complex roots. We describe some low level techniques for speeding up the algorithm. Its performance is compared with the well-known MPSolve library and Maple.

References

  1. 1.
    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, pp. 71–78. ACM (2016)Google Scholar
  2. 2.
    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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bini, D.A., Fiorentino, G.: Design, analysis, and implementation of a multiprecision polynomial rootfinder. Numer. Algorithms 23(2–3), 127–173 (2000)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bini, D.A., Robol, L.: Solving secular and polynomial equations: a multiprecision algorithm. J. Comput. Appl. Math. 272, 276–292 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brönnimann, H., Burnikel, C., Pion, S.: Interval arithmetic yields efficient dynamic filters for computational geometry. Discrete Appl. Math. 109(1–2), 25–47 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Emiris, I.Z., Pan, V.Y., Tsigaridas, E.P.: Algebraic algorithms. In: Computing Handbook, Third Edition: Computer Science and Software Engineering, pp. 10:1–10:30. Chapman and Hall/CRC (2014)Google Scholar
  7. 7.
    Fortune, S.: An iterated eigenvalue algorithm for approximating roots of univariate polynomials. J. Symb. Comput. 33(5), 627–646 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Giusti, M., Lecerf, G., Salvy, B., Yakoubsohn, J.-C.: On location and approximation of clusters of zeros of analytic functions. Found. Comput. Math. 5(3), 257–311 (2005)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gourdon, X.: Combinatoire, Algorithmique et Géométrie des Polynomes. Ph.D. thesis, École Polytechnique (1996)Google Scholar
  10. 10.
    Hribernig, V., Stetter, H.J.: Detection and validation of clusters of polynomial zeros. J. Symb. Comput. 24(6), 667–681 (1997)MathSciNetCrossRefGoogle Scholar
  11. 11.
    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, pp. 303–310. ACM (2016)Google Scholar
  12. 12.
    Niu, X.-M., Sakurai, T., Sugiura, H.: A verified method for bounding clusters of zeros of analytic functions. J. Comput. Appl. Math. 199(2), 263–270 (2007)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pan, V.Y.: Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding. J. Symb. Comput. 33(5), 701–733 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Rouillier, F., Zimmermann, P.: Efficient isolation of polynomial’s real roots. J. Comput. Appl. Math. 162(1), 33–50 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Sagraloff, M., Mehlhorn, K.: Computing real roots of real polynomials. J. Symb. Comput. 73, 46–86 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Yap, C., Sagraloff, M., Sharma, V.: Analytic root clustering: a complete algorithm using soft zero tests. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds.) CiE 2013. LNCS, vol. 7921, pp. 434–444. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-39053-1_51CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.TU KaiserslauternKaiserslauternGermany
  2. 2.City University of New YorkNew YorkUSA
  3. 3.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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