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- Newton_fractal abstract "The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial . It is the Julia set of the meromorphic function which is given by Newton's method. When there are no attractive cycles (of order greater than 1), it divides the complex plane into regions , each of which is associated with a root of the polynomial, . In this way the Newton fractal is similar to the Mandelbrot set, and like other fractals it exhibits an intricate appearance arising from a simple description. It is relevant to numerical analysis because it shows that (outside the region of quadratic convergence) the Newton method can be very sensitive to its choice of start point.Many points of the complex plane are associated with one of the roots of the polynomial in the following way: the point is used as starting value for Newton's iteration , yielding a sequence of points , , .... If the sequence converges to the root , then was an element of the region . However, for every polynomial of degree at least 2 there are points for which the Newton iteration does not converge to any root: examples are the boundaries of the basins of attraction of the various roots. There are even polynomials for which open sets of starting points fail to converge to any root: a simple example is , where some points are attracted by the cycle 0, 1, 0, 1 ... rather than by a root.An open set for which the iterations converge towards a given root or cycle (that is not a fixed point), is a Fatou set for the iteration. The complementary set to the union of all these, is the Julia set. The Fatou sets have common boundary, namely the Julia set. Therefore each point of the Julia set is a point of accumulation for each of the Fatou sets. It is this property that causes the fractal structure of the Julia set (when the degree of the polynomial is larger than 2).To plot interesting pictures, one may first choose a specified number of complex points and compute the coefficients of the polynomial.Then for a rectangular lattice , , ..., , , ..., of points in , one finds the index of the corresponding root and uses this to fill an ×raster grid by assigning to each point a colour . Additionally or alternatively the colours may be dependent on the distance , which is defined to be the first value such that for some previously fixed small .".
- Newton_fractal thumbnail Julia_set_for_the_rational_function.png?width=300.
- Newton_fractal wikiPageExternalLink 313081.html.
- Newton_fractal wikiPageExternalLink thesis.pl?thesis06-1.
- Newton_fractal wikiPageExternalLink ~scott.
- Newton_fractal wikiPageID "3918520".
- Newton_fractal wikiPageRevisionID "596926073".
- Newton_fractal hasPhotoCollection Newton_fractal.
- Newton_fractal subject Category:Fractals.
- Newton_fractal subject Category:Numerical_analysis.
- Newton_fractal type Abstraction100002137.
- Newton_fractal type Cognition100023271.
- Newton_fractal type Form105930736.
- Newton_fractal type Fractal105931152.
- Newton_fractal type Fractals.
- Newton_fractal type PsychologicalFeature100023100.
- Newton_fractal type Structure105726345.
- Newton_fractal comment "The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial . It is the Julia set of the meromorphic function which is given by Newton's method. When there are no attractive cycles (of order greater than 1), it divides the complex plane into regions , each of which is associated with a root of the polynomial, .".
- Newton_fractal label "Fractale de Newton".
- Newton_fractal label "Newton fractal".
- Newton_fractal label "Newton-Fraktal".
- Newton_fractal label "Wstęga Newtona".
- Newton_fractal label "Бассейны Ньютона".
- Newton_fractal label "كسيرية نيوتن".
- Newton_fractal sameAs Fraktál_Newton.
- Newton_fractal sameAs Newton-Fraktal.
- Newton_fractal sameAs Fractale_de_Newton.
- Newton_fractal sameAs Wstęga_Newtona.
- Newton_fractal sameAs m.0b6r5z.
- Newton_fractal sameAs Q1151001.
- Newton_fractal sameAs Q1151001.
- Newton_fractal sameAs Newton_fractal.
- Newton_fractal wasDerivedFrom Newton_fractal?oldid=596926073.
- Newton_fractal depiction Julia_set_for_the_rational_function.png.
- Newton_fractal isPrimaryTopicOf Newton_fractal.
