Elusive Zeros under Newton’s Method

Trevor M. O’Brien, Gareth E. Roberts

Department of Computer Science, Brown University, Providence, USA.
Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, USA.

DOI: 10.4236/am.2014.515231                

Abstract

Though well-known for its simplicity and efficiency, Newton’s method applied to a complex polynomial can fail quite miserably, even on a relatively large open set of initial guesses. In this work, we present some analytic and numerical results for Newton’s method applied to the complex quartic family where is a parameter. The symmetric location of the roots of allows for some easy reductions. In particular, when λ is either real or purely imaginary, standard techniques from real dynamical systems theory can be employed for rigorous analysis. Classifying those λ-values where Newton’s method fails on an open set leads to complex and aesthetically intriguing geometry in the λ-parameter plane, complete with fractal-like figures such as Mandelbrot-like sets, tricorns and swallows.

Keywords

Newton’s Method, Complex Dynamical Systems, Mandelbrot-Like Sets, Tricorns

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Curry, J.H., Garnett, L. and Sullivan, D. (1983) On the Iteration of a Rational Function: Computer Experiments with Newton’s Method. Communications in Mathematical Physics, 91, 267-277. http://dx.doi.org/10.1007/BF01211162
[2]	Blanchard, P. (1994) The Dynamics of Newton’s Method. Complex Dynamical Systems, Cincinnati. Proceedings of Symposia in Applied Mathematics, Vol. 49, AMS, Providence, 139-154.
[3]	Head, J.E. (1988) The Combinatorics of Newton’s Method for Cubic Polynomials. Doctoral Dissertation, Cornell University, Ithaca.
[4]	Lei, T. (1990) Cubic Newton’s Method of Thurston’s Type. Laboratoire de Mathématiques, Ecole Normale Superieure de Lyon. Preprint.
[5]	Roberts, G.E. and Horgan-Kobelski, J. (2004) Newton’s versus Halley’s Method: A Dynamical Systems Approach. International Journal of Bifurcation and Chaos, 14, 3459-3475. http://dx.doi.org/10.1142/S0218127404011399
[6]	Haeseler, F.V. and Kriete, H. (1993) Surgery for Relaxed Newton’s Method. Complex Variables, Theory and Application, 22, 129-143. http://dx.doi.org/10.1080/17476939308814653
[7]	Douady, A. and Hubbard, J.H. (1985) On the Dynamics of Polynomial-Like Mappings. Annales Scientifiques de L’Ecole Normal Superieure, 4e serie, t. 18, 287-343.
[8]	Milnor, J. (1992) Remarks on Iterated Cubic Maps. Experimental Mathematics, 1, 5-24.
[9]	Blanchard, P. (1981) Complex Analytic Dynamics on the Riemann Sphere. Bulletin of the American Mathematical Society (New Series), 11, 85-141. http://dx.doi.org/10.1090/S0273-0979-1984-15240-6
[10]	Milnor, J. (2006) Dynamics in One Complex Variable. 3rd Edition, Princeton University Press, Princeton.
[11]	Sutherland, S. (1989) Finding Roots of Complex Polynomials with Newton’s Method. Doctoral Dissertation, Boston University, Boston.
[12]	Devaney, R.L. (1992) A First Course in Chaotic Dynamical Systems. Westview Press.
[13]	MAPLE, Version 15.00 (2011) Maplesoft. Waterloo Maple Inc., Waterloo.