ISSN:
1432-0916
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
,
Physics
Notes:
Abstract Using Newton's method to look for roots of a polynomial in the complex plane amounts to iterating a certain rational function. This article describes the behavior of Newton iteration for cubic polynomials. After a change of variables, these polynomials can be parametrized by a single complex parameter, and the Newton transformation has a single critical point other than its fixed points at the roots of the polynomial. We describe the behavior of the orbit of the free critical point as the parameter is varied. The Julia set, points where Newton's method fail to converge, is also pictured. These sets exhibit an unexpected stability of their gross structure while the changes in small scale structure are intricate and subtle.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01211162
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