Fractal Basins of Attraction Associated with a Damped Newton's Method

Details Fractal Basins of Attraction Associated with a Damped Newton's Method Fractal Basins of Attraction Associated with a Damped Newton's Method

1996-12-05

arxiv.org/abs/chao-dyn/9612010v1

Nonlinear Sciences

Chaotic Dynamics

11 pages of LateX output with five 8-bit color .ps figures. To appear in SIAM Review, 1997

Scientific paper

An intriguing and unexpected result for students learning numerical analysis is that Newton's method, applied to the simple polynomial z^3 - 1 = 0 in the complex plane, leads to intricately interwoven basins of attraction of the roots. As an example of an interesting open question that may help to stimulate student interest in numerical analysis, we investigate the question of whether a damping method, which is designed to increase the likelihood of convergence for Newton's method, modifies the fractal structure of the basin boundaries. The overlap of the frontiers of numerical analysis and nonlinear dynamics provides many other problems that can help to make numerical analysis courses interesting.

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