Abstract
Given a complex polynomial p(z) with at least three distinct roots, we first prove that no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove that the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, B m (z), m≥2. Let θ be a simple root of p(z), V(θ) its Voronoi cell, and A m (θ) its basin of attraction with respect to B m (z). We prove that given any closed subset C of V(θ), including any homothetic copy of V(θ), there exists m 0 such that for all m≥m 0, C is also a subset of A m (θ). This implies that when all roots of p(z) are simple, the basins of attraction of B m (z) uniformly approximate the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of B m (z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of the Voronoi diagram. In a sense, this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry.
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References
Asano, T., Matoušek, J., Tokuyama, T.: Zone diagrams: existence, uniqueness, and algorithmic challenge. SIAM J. Comput. 37, 1182–1198 (2007)
Aurenhammer, F.: Voronoi diagrams—a survey of fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991)
Beardon, A.F.: Iteration of Rational Functions: Complex Analytic Dynamical Systems. Springer, New York (1991)
Cayley, A.: The Newton–Fourier imaginary problem. Am. J. Math. 2, 97 (1879)
de Biasi, S.C., Kalantari, B., Kalantari, I.: Maximal zone diagrams and their computation. In: Proceedings of the Seventh Annual International Symposium on Voronoi Diagrams in Science and Engineering, pp. 171–180 (2010)
de Biasi, S.C., Kalantari, B., Kalantari, I.: Mollified zone diagrams and their computation (2011, in preparation)
Goodman, J.E., O’Rourke, J. (eds.): Handbook of Discrete and Computational Geometry, 2nd edn. Discrete Mathematics and Its Applications. Chapman & Hall, Boca Raton (2004)
Kalantari, B.: Generalization of Taylor’s theorem and Newton’s method via a new family of determinantal interpolation formulas and its applications. J. Comput. Appl. Math. 126, 287–318 (2000)
Kalantari, B.: On homogeneous linear recurrence relations and approximation of zeros of complex polynomials. In: DIMACS Series in Discrete Mathematics and Theoretical Computer Science. Unusual Applications in Number Theory, vol. 64, pp. 125–143 (2004)
Kalantari, B.: Polynomial Root-Finding and Polynomiography. World Scientific, New Jersey (2008)
Kalantari, B.: Voronoi diagrams and polynomial root-finding. In: Proceedings of the Sixth Annual International Symposium on Voronoi Diagrams in Science and Engineering, pp. 31–40 (2009)
Kalantari, B., Jin, Y.: On extraneous fixed-points of the basic family of iteration functions. BIT 43, 453–458 (2003)
Kalantari, B., Pate, T.H.: A Determinantal lower bound. Linear Algebra Appl. 326, 151–159 (2001)
McMullen, C.: Families of rational maps and iterative root-finding algorithms. Ann. Math. 125, 467–493 (1987)
Milnor, J.: Dynamics in One Complex Variable: Introductory Lectures, vol. 160, 3rd edn. Princeton University Press, New Jersey (2006)
Preparata, F.P., Shamos, M.I.: Computational Geometry: An Introduction. Springer, New York (1985)
Schröder, E.: On infinitely many algorithms for solving equations. Math. Ann. 2, 317–365 (1870) (German). (English translation by G.W. Stewart, TR-92-121, Institute for Advanced Computer Studies, University of Maryland, College Park, MD, 1992.)
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice Hall, Englewood Cliffs (1964)
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Kalantari, B. Polynomial Root-Finding Methods Whose Basins of Attraction Approximate Voronoi Diagram. Discrete Comput Geom 46, 187–203 (2011). https://doi.org/10.1007/s00454-011-9330-3
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DOI: https://doi.org/10.1007/s00454-011-9330-3