Newton fractal

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The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial $p(Z)\in\mathbb{C}[Z]$ . When there are no attractive cycles, it divides the complex plane into regions $G k$ , each of which is associated with a root $ζ k$ of the polynomial, $k=1,..,\operatorname{deg}(p)$ . In this way the Newton fractal is similar to the Mandelbrot set, and like other fractals it exhibits a complex 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 $\operatorname{deg}(p)$ roots of the polynomial in the following way: the point is used as starting value $z 0$ for Newton's iteration $z_{n+1}:=z_n-\frac{p(z)}{p'(z)}$ , yielding a sequence of points $z 1$ , $z 2$ , .... If the sequence converges to the root ζ_k, then z₀ was an element of the region G_k. 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 z³-2z+2, where some points are attracted by the cycle 0, 1, 0, 1 ... rather than by a root.

To plot interesting pictures, one may first choose a specified number d of complex points (ζ₁,...,ζ_d) and compute the coëfficients (p₁,...,p_d) of the polynomial

$p(z)=z^d+p_1z^{d-1}+\cdots+p_{d-1}z+p_d:=(z-\zeta_1)\cdot\cdots\cdot(z-\zeta_d)$ .

Then for a rectangular lattice z_mn = z₀₀ + mΔx + inΔy, m = 0, ..., M - 1, n = 0, ..., N - 1 of points in $\mathbb{C}$ , one finds the index k(m,n) of the corresponding root ζ_k(m,n) and uses this to fill an M×N raster grid by assigning to each point (m,n) a colour f_k(m,n). Additionally or alternatively the colours may be dependent on the distance D(m,n), which is defined to be the first value D such that |z_D - ζ_k(m,n)| < ε for some previously fixed small ε > 0.

[edit] Generalization of Newton fractals

A generalization of Newton's iteration is

$z_{n+1}=z_n- a \frac{p(z_n)}{p'(z_n)}$

where a is any complex number^[1]. The special choice $a = 1$ corresponds to the Newton fractal. The fixed points of this map are stable when $a$ lies inside the disk of radius 1 centered at 1. When a 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 p is a polynomial of degree n, then the sequence z_n is bounded provided that a is inside a disk of radius n centered at n.

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

Newton fractal for three degree-3 roots ( $p (z) = z 3 - 1$ ), coloured by number of iterations required	Newton fractal for three degree-3 roots ( $p (z) = z 3 - 1$ ), coloured by root reached	Newton fractal for $p (z) = z 3 - 2 z + 2$ . Points in the red basins do not reach a root.	Newton fractal for $x 8 + 15 x 4 - 16$
Newton fractal for $p (z) = z 5 - 3 i z 3 - (5 + 2 i) z 2 + 3 z + 1$ , coloured by root reached, shaded by number of iterations required	Newton fractal for $p (z) = s i n (z)$ , coloured by root reached, shaded by number of iterations required	Another Newton fractal for $s i n (x)$	Generalized Newton fractal for $p (z) = z 3 - 1$ , $a = - 0.5.$ The colour was chosen based on the argument after 40 iterations.
Generalized Newton fractal for $p (z) = z 2 - 1$ , $a = 1 + i .$	Generalized Newton fractal for $p (z) = z 3 - 1$ , $a = 2.$	Generalized Newton fractal for $p (z) = z 4 + 3 i - 1$ , $a = 2.1.$

[edit] See also

[edit] References

^ Simon Tatham. Fractals derived from Newton-Raphson.

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
On the Number of Iterations for Newton's Method by Dierk Schleicher July 21, 2000
Newton's Method as a Dynamical System by Johannes Rueckert