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Elusive Zeros under Newton’s Method

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DOI: 10.4236/am.2014.515231    3,892 Downloads   4,361 Views  

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.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

O’Brien, T. and Roberts, G. (2014) Elusive Zeros under Newton’s Method. Applied Mathematics, 5, 2393-2407. doi: 10.4236/am.2014.515231.

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