Elusive Zeros under Newton’s Method

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.