Elusive Zeros under Newton’s Method
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.