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Properties

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	~~[[File:Julia set for the rational function.png\|240px\|right\|thumb\|Julia set for the rational function associated to Newton's method for ƒ:z→z<sup>3</sup>−1.]]~~		Hello, my title is Andrew and my wife doesn't like it at all. To climb is something I really appreciate doing. Ohio is exactly where my house is but my husband desires us to transfer. Invoicing is what I do for a living but I've usually wanted my own business.<br><br>My homepage: psychic chat online ([http://netwk.hannam.ac.kr/xe/data_2/85669 just click netwk.hannam.ac.kr])

	The '''Newton fractal''' is a [[boundary set]] in the [[complex plane]] which is characterized by [[Newton's method]] applied to a fixed [[polynomial]] <math>p(Z)\in\mathbb{C}[Z]</math>. It is the [[Julia set]] of the [[meromorphic function]] <math>z\mapsto z-\tfrac{p(z)}{p'(z)}</math> 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 <math>G_k</math>, each of which is associated with a [[root of a function\|root]] <math>\zeta_k</math> of the polynomial, <math>k=1,..,\operatorname{deg}(p)</math>. 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 <math>\operatorname{deg}(p)</math> roots of the polynomial in the following way: the point is used as starting value <math>z_0</math> for Newton's iteration <math>z_{n+1}:=z_n-\frac{p(z_n)}{p'(z_n)}</math>, yielding a sequence of points <math>z_1</math>, <math>z_2</math>, .... If the sequence converges to the root <math>\zeta_k</math>, then <math>z_0</math> was an element of the region <math>G_k</math>. 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 <math>z^3-2z+2</math>, 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 <math>d</math> of complex points <math>(\zeta_1,...,\zeta_d)</math> and compute the coefficients <math>(p_1,...,p_d)</math> of the polynomial
	~~:<math>p(z)=z^d+p_1z^{d-1}+\cdots+p_{d-1}z+p_d:=(z-\zeta_1)\cdot\cdots\cdot(z-\zeta_d)</math>.~~
	Then for a rectangular lattice <math>z_{mn} = z_{00} + m \Delta x + in \Delta y</math>, <math>m = 0</math>, ..., <math>M - 1</math>, <math>n = 0</math>, ..., <math>N - 1</math> of points in <math>\mathbb{C}</math>, one finds the index <math>k(m,n)</math> of the corresponding root <math>\zeta_{k(m,n)}</math> and uses this to fill an <math>M</math>×<math>N</math> raster grid by assigning to each point <math>(m,n)</math> a colour <math>f_{k(m,n)}</math>. Additionally or alternatively the colours may be dependent on the distance <math>D(m,n)</math>, which is ~~defined to be the first value <math>D</math> such that <math>\|z_D - \zeta_{k(m,n)}\| < \epsilon</math> for some previously fixed small <math>\epsilon > 0</math>~~.

	~~== Generalization of Newton fractals ==~~

	~~A generalization of Newton's iteration~~ is
	~~:<math>z_{n+1}=z_n- a \frac{p(z_n)}{p'(z_n)} </math>~~
	where ~~<math>a</math>~~ is any complex number.<ref>{{cite web \| url = http://www.chiark.greenend.org.uk/~sgtatham/newton/ \| title = Fractals derived from Newton-Raphson \| author = Simon Tatham }}</ref> The special choice <math>a=1</math> corresponds to ~~the Newton fractal.~~
	The fixed points of this map are stable when <math>a</math> lies inside the disk of radius 1 centered at 1. When <math>a</math> is outside this disk, the fixed points are locally unstable, however the map still exhibits a fractal structure in the sense of [[Julia set]]. If <math>p</math> is a polynomial of degree <math>n</math>, then the sequence <math>z_n</math> is [[bounded]] provided that <math>a</math> is inside a disk of radius <math>n</math> centered at <math>n</math>.

	~~More generally, Newton's fractal~~ is ~~a special case of a [[Julia set]].~~

	~~<gallery>~~
	~~Image:FRACT008.png\|Newton fractal for three degree-3 roots (<math>p(z)=z^3-1</math>), coloured by number of iterations required~~
	~~Image:Newtroot 1 0 0 m1.png\|Newton fractal for three degree-3 roots (<math>p(z)=z^3-1</math>), coloured by root reached~~
	~~Image:Newton_z3-2z+2.png\|Newton fractal for <math>p(z)=z^3-2z+2</math>. Points in the red basins~~ do ~~not reach a root.~~
	~~Image:Colored Newton Fractal 2.png\|Newton fractal~~ for a ~~7th order polynomial, colored by root reached and shaded by rate of convergence.~~
	~~Image:timelapse34.jpg\|Newton fractal for <math>x^8+15x^4-16</math>~~
	~~Image:Newtroot 1 0 m3i m5m2i 3 1.png\|Newton fractal for <math>p(z)=z^5-3iz^3-(5+2i)z^2+3z+1</math>, coloured by root reached, shaded by number of iterations required.~~
	~~Image:timelapse4.jpg\|Newton fractal for <math>p(z)=\sin(z)</math>, coloured by root reached, shaded by number of iterations required~~
	~~Image:Sin(x)_detail.png\|Another Newton fractal for <math>\sin(x)</math>~~
	~~Image:Mnfrac1.png\| Generalized Newton fractal for <math>p(z)=z^3-1</math>, <math>a=-0.5.</math> The colour was chosen based on the argument after 40 iterations.~~
	~~Image:Mnfrac2.png\| Generalized Newton fractal for <math>p(z)=z^2-1</math>, <math> a=1+i.</math>~~
	~~Image:Mnfrac3.png\| Generalized Newton fractal for <math>p(z)=z^3-1</math>, <math>a=2.</math>~~
	~~Image:Mnfrac4.png\| Generalized Newton fractal for <math>p(z)=z^{4+3i}-1</math>, <math>a=2.1.</math>~~
	~~Image:Newton z6 z3.jmb.jpg\|<center><math>p(z) = z^6 + z^3 - 1</math></center>~~
	~~Image:Newton SINUS.jmb~~.~~jpg\|~~<~~center~~><~~math>p(z) = \sin(z)- 1</math></center~~>
	~~Image~~:~~Newton COSH.jmb.jpg\|<center><math>p~~(~~z) = \cosh(z)- 1</math></center>~~
	~~</gallery>~~

	~~== See also ==~~
	* [[Julia set]]
	* [[Mandelbrot set]]
	* [[Nova fractal]]

	~~==References==~~
	~~{{reflist}}~~
	* [http://~~www~~.~~math~~.~~sunysb~~.~~edu~~/~~~scott~~/ J. H. ~~Hubbard, D~~. ~~Schleicher, S. Sutherland~~]~~: ''How to Find All Roots of Complex Polynomials by Newton's Method'', Inventiones Mathematicae vol. 146 (2001~~) ~~- with a discussion of the global structure of Newton fractals~~
	*[http://citeseer.ist.psu.edu/313081.html ''On the Number of Iterations for Newton's Method''] by Dierk Schleicher July 21, 2000
	*[http://www.math.sunysb.edu/cgi-bin/thesis.pl?thesis06-1 ''Newton's Method as a Dynamical System''] by Johannes Rueckert
	~~{{Commons\|Newton fractal\|Newton fractal}}~~

	~~[[Category:Numerical analysis]]~~
	~~[[Category:Fractals]]~~

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