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USB1 - Method of creating graphical works based on polynomials - Google Patents

A certain value may not even converge. This leads to areas that are called basins of attraction. Values within the basin converge quickly to the root.


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The interesting things happen at the edges of the basin, where a small shift in value creates a large change in the convergence of the root. The basic process of making a fractal image involves guessing a starting point and using an algorithm to find a better point.


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This process is done recursively until the value is found within a certain buffer size, or after a certain number of iterations. When a starting point is picked, one hopes that the iterations will lead to the correct root. However, this is not always the case and this leads to the basis of fractals.

ISBN 10: 9812700595

A polynomiograph of a degree 36 polynomial [i]. Fractals are related to polynomials through a branch of mathematics called polynomiography. A good deal of work has been done on the process of finding the roots of small to moderate degree polynomials, but not much work has been done on finding the roots to large polynomials because large polynomials do not occur very frequently in mathematical studies. In addition, the larger the polynomial, the more difficult it becomes to find its roots. Polynomiography changes all of this by using an algorithm to estimate the roots of the polynomial graphically.

Polynomial Root-finding and Polynomiography

As a result, it has become much easier to approximate and find the roots of large polynomials. Igor Potapov Department of Computer Science.

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Rational Roots Test / Theorem

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