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http://www.scirp.org/journal/PaperInformation.aspx?PaperID=48889#.VDc0L1fHRK0
Author(s)
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.
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.
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