Elusive Zeros under Newton’s Method

Affiliation(s)

Department of Computer Science, Brown University, Providence, USA.

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

Department of Computer Science, Brown University, Providence, USA.

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

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.

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.

O’Brien, T. and Roberts, G. (2014) Elusive Zeros under Newton’s Method.

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