Numerical Analysis Group
Research Report NA-96/14
Fractal Characteristics of Newton's Method on Polynomials
Michael Drexler,
Ian Sobey and
C. Bracher
November 1996, 61 pages.
In this report, we present a simple geometric generation principle for
the fractal that is obtained when applying Newton's method to find the roots of
a general complex polynomial with real coefficients. For the case of symmetric
polynomials
, the generation mechanism is derived from first principles.
We discuss the case of a general cubic and are able to give a description of the
arising fractal structure depending on the coefficients of the cubic. Special
cases are analysed and their characteristics, including scale factors and an
approximate fractal dimension, are derived. The theoretical results are confirmed
via computational experiments. An application of the theory in turbulence modelling
is presented.
- Subject classifications:
- not yet defined
- Key words and phrases:
- Newton's Methods, Fractals, Iterative mappings, Polynomials
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