Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE DISTANCE TO A ROOT OF COMPLEX POLYNOMIALS UNDER NEWTON'S METHOD

  • Chaiya, Malinee (Department of Mathematics Faculty of Science Silpakorn University) ;
  • Chaiya, Somjate (Department of Mathematics Faculty of Science Silpakorn University)
  • Received : 2019.07.01
  • Accepted : 2020.07.07
  • Published : 2020.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we derive an upper bound for the distance from a point in the immediate basin of a root of a complex polynomial to the root itself. We establish that if z is a point in the immediate basin of a root α of a polynomial p of degree d ≥ 12, then ${\mid}z-{\alpha}{\mid}{\leq}{\frac{3}{\sqrt{d}}\(6{\sqrt{310}}/35\)^d{\mid}N_p(z)-z{\mid}$, where Np is the Newton map induced by p. This bound leads to a new bound of the expected total number of iterations of Newton's method required to reach all roots of every polynomial p within a given precision, where p is normalized so that its roots are in the unit disk.

Keywords

Acknowledgement

We would like to express our sincere gratitude to the referees for their valuable suggestions and comments which improve the paper.

References

  1. T. Bilarev, M. Aspenberg, and D. Schleicher, On the speed of convergence of Newton's method for complex polynomials, Math. Comp. 85 (2016), no. 298, 693-705. https://doi.org/10.1090/mcom/2985
  2. S. Chaiya, On the distance to a root of polynomials, Abstr. Appl. Anal. 2011 (2011), Art. ID 495312, 6 pp. https://doi.org/10.1155/2011/495312
  3. J. Hubbard, D. Schleicher, and S. Sutherland, How to find all roots of complex polynomials by Newton's method, Invent. Math. 146 (2001), no. 1, 1-33. https://doi.org/10.1007/s002220100149
  4. F. Przytycki, Remarks on the simple connectedness of basins of sinks for iterations of rational maps, in Dynamical systems and ergodic theory (Warsaw, 1986), 229-235, Banach Center Publ., 23, PWN, Warsaw, 1989.
  5. D. Schleicher, On the number of iterations of Newton's method for complex polynomials, Ergodic Theory Dynam. Systems 22 (2002), no. 3, 935-945. https://doi.org/10.1017/S0143385702000482
  6. D. Schleicher, On the efficient global dynamics of Newton's method for complex polynomials, Manuscript (2016), arXiv:1108.5773v3.
  7. D. Schleicher and R. Stoll, Newton's method in practice: Finding all roots of polynomials of degree one million efficiently, Theoret. Comput. Sci. 681 (2017), 146-166. https://doi.org/10.1016/j.tcs.2017.03.025
  8. M. Shishikura, The connectivity of the Julia set and fixed points, in Complex dynamics, 257-276, A K Peters, Wellesley, MA, 2009.