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Counting Roots of a Polynomial in a Convex Compact Region by Means of Winding Number Calculation via Sampling

  • Vitaly Zaderman
  • Liang ZhaoEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11661)

Abstract

In this paper, we propose a novel efficient algorithm for calculating winding numbers, aiming at counting the number of roots of a given polynomial in a convex region on the complex plane. This algorithm can be used for counting and exclusion tests in a subdivision algorithms for polynomial root-finding, and would be especially useful in application scenarios where high-precision polynomial coefficients are hard to obtain but we succeed with counting already by using polynomial evaluation with lower precision. We provide the pseudo code of the algorithm as well as a proof of its correctness.

Keywords

Polynomial root-finding Winding number 

Notes

Acknowledgements

Our research has been supported by the NSF Grant CCF–1563942, NSF Grant CCF-1733834, and the PSC CUNY Award 69813 00 48.

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.
    Henrici, P.: Applied and Computational Complex Analysis. vol. 1, Power Series, Integration, Conformal Mapping, Location of Zeros. John Wiley, New York (1974)zbMATHGoogle Scholar
  4. 4.
    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
  5. 5.
    Householder, A.S.: Dandelin, Lobačevskii, or Graeffe? Am. Math. Mon. 66(6), 464–466 (1959)zbMATHGoogle Scholar
  6. 6.
    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_28CrossRefGoogle Scholar
  7. 7.
    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
  8. 8.
    Renegar, J.: On the worst-case arithmetic complexity of approximating zeros of polynomials. J. Complex. 3(2), 90–113 (1987)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Schönhage, A.: The fundamental theorem of algebra in terms of computational complexity. Manuscript. Univ. of Tübingen, Germany (1982)Google Scholar
  10. 10.
    Weyl, H.: Randbemerkungen zu hauptproblem der mathematik. Mathematische Zeitschrift 20, 131–150 (1924)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Zapata, J.L.G., Martín, J.C.D.: A geometric algorithm for winding number computation with complexity analysis. J. Complex. 28(3), 320–345 (2012)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Zapata, J.L.G., Martín, J.C.D.: Finding the number of roots of a polynomial in a plane region using the winding number. Comput. Math. Appl. 67(3), 555–568 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Ph.D. Program in MathematicsThe Graduate Center of the City University of New YorkNew YorkUSA
  2. 2.Department of Computer ScienceThe Graduate Center of the City University of New YorkNew YorkUSA

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