Elusive Zeros under Newton’s Method

• 作者: Gareth E. Roberts Trevor M. O’Brien
• 来源期刊: 应用数学(英文)
• 学科: 数学
• 关键词: Newton’s Method COMPLEX DYNAMICAL SYSTEMS Mandelbrot-Like SETS Tricorns
• 年，卷（期）: 2014,（15）
• 页码范围: 2393-2407
• 页数: 15页
• 分类号: O1

摘要

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.