On Koenig's theorem for integer functions of finite order


  • A.N. Gromov Moscow State Institute of International Relations at Odintsovo




logarithmic derivative, higher-order derivative, simplest fractions, convergence radius of power series, Voronoi polygons (cells), global convergence


It is shown that Koenig's theorem on zeros of analytic functions applied to the logarithmic derivative of an integer function of finite order leads to an algorithm of finding zeros whose convergence domains are the Voronoi polygons of the zeros to be found. Since the Voronoi diagram of a sequence of zeros is a set of measure zero, this algorithm is globally convergent. The rate of convergence is estimated. For higher-order iterations that are constructed using Koenig's theorem, the effect of root multiplicity on the convergence domain is considered and the convergence rate is estimated for this case.

Author Biography

A.N. Gromov

MGIMO University,
• Senior Lecturer


  1. F. P. Preparata and M. I. Shamos, Computational Geometry: An Introduction (Springer, New York, 1985; Mir, Moscow, 1989).
  2. I. S. Berezin and N. P. Zhidkov, Computing Methods (Nauka, Moscow, 1962; Oxford, Pergamon, 1965).
  3. A. I. Markushevich, The Theory of Analytic Functions (Nauka, Moscow, 1967; Chelsea, New York, 1977).
  4. A. N. Gromov, “An Approach for Constructing One-Point Iterative Methods for Solving Nonlinear Equations of One Variable,” Vychisl. Metody Programm. 16, 298-306 (2015).
  5. A. N. Gromov, “A Globally Convergent Method for Finding Zeros of Integer Functions of Finite Order,” Vychisl. Metody Programm. 18, 115-128 (2017).
  6. P. D. Proinov and S. I. Ivanov, “Convergence Analysis of Sakurai-Torii-Sugiura Iterative Method for Simultaneous Approximation of Polynomial Zeros,” J. Comput. Appl. Mat. 357, 56-70 (2019).
  7. H. Sugiura and T. Hasegawa, “On the Global Convergence of Schröder’s Iteration Formula for Real Zeros of Entire Functions,” J. Comput. Appl. Math. 358, 136-145 (2019).
  8. J. M. Gutiérrez and M. Á. Hernández-Verón, “An Acceleration of the Continuous Newton’s Method,” J. Comput. Appl. Math. 354, 213-220 (2019).
  9. J. L. García-Zapata, J. C. D. Martín, and Á. C. Fácila, “An Adaptive Subdivision Method for Root Finding of Univariate Polynomials,” J. Comput. Appl. Math. 352, 146-164 (2019).
  10. M. Lázaro, P. Martín, A. Agüero, and I. Ferrer, “The Polynomial Pivots as Initial Values for a New Root-Finding Iterative Method,” J. Appl. Math. 2015 (2015).
    doi 10.1155/2015/413816



How to Cite

Громов А. On Koenig’s Theorem for Integer Functions of Finite Order // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2020. 21. 280-289. doi 10.26089/NumMet.v21r324



Methods and algorithms of computational mathematics and their applications