Newton fractal

 ( Numerical Analysis ) 

The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial $\backslash mathbb\{C\}$ or transcendental function. 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.

Almost all points of the complex plane are associated with one of the roots of a given 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 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 images of the fractal, one may first choose a specified number of complex points and compute the coefficients of the polynomial

$p(z)=z^d+p\_1z^\{d-1\}+\backslash cdots+p\_\{d-1\}z+p\_d:=(z-\backslash zeta\_1)(z-\backslash zeta\_2)\backslash cdots(z-\backslash zeta\_d)$.

$z\_\{mn\}\; =\; z\_\{00\}\; +\; m\; \backslash ,\; \backslash Delta\; x\; +\; in\; \backslash ,\; \backslash Delta\; y;\; \backslash quad\; m\; =\; 0,\; \backslash ldots,\; M\; -\; 1;\; \backslash quad\; n\; =\; 0,\; \backslash ldots,\; N\; -\; 1$

$z\_\{n+1\}=z\_n-\; a\; \backslash frac\{p(z\_n)\}\{p\text{'}(z\_n)\}$

where is any complex number. The special choice corresponds to the Newton fractal. The fixed points of this map are stable when lies inside the disk of radius 1 centered at 1. When 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 is a polynomial of degree , then the sequence is Bounded set provided that is inside a disk of radius centered at .

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

$z\_\{n+1\}=z\_n-\; a\; \backslash frac\{p(z\_n)\}\{p\text{'}(z\_n)\}\; +\; c\; =\; G(a,\; c,\; z)$

The "Julia" variant of the Nova fractal keeps constant over the image and initializes to the pixel coordinates. The "Mandelbrot" variant of the Nova fractal initializes to the pixel coordinates and sets to a critical point, where

$\backslash frac\{\backslash partial\}\{\backslash partial\; z\}\; G(a,\; c,\; z)\; =\; 0.$

Commonly-used polynomials like or lead to a critical point at .

The three roots of the function are

$z\; =\; 1,\backslash \; -\backslash tfrac12\; +\; \backslash tfrac\{\backslash sqrt\; 3\}\{2\}i,\backslash \; -\backslash tfrac12\; -\; \backslash tfrac\{\backslash sqrt\; 3\}\{2\}i$

The above-defined functions can be translated in pseudocode as follows: // z^3-1 float2 Function (float2 z) {

return cpow(z, 3) - float2(1, 0); // cpow is an exponential function for complex numbers
     

// 3*z^2 float2 Derivative (float2 z) {

return 3 * cmul(z, z); // cmul is a function that handles multiplication of complex numbers
     

float2(1, 0),
float2(-.5, sqrt(3)/2),
float2(-.5, -sqrt(3)/2)
     

   red,
   green,
   blue
     

zx = scaled x coordinate of pixel (scaled to lie in the Mandelbrot X scale (-2.5, 1))
   zy = scaled y coordinate of pixel (scaled to lie in the Mandelbrot Y scale (-2, 1))
     

   float2 z = float2(zx, zy); // z is originally set to the pixel coordinates
     

for (int iteration = 0;
     iteration < maxIteration;
     iteration++;)
{
	z -= cdiv(Function(z), Derivative(z)); // cdiv is a function for dividing complex numbers
     

       float tolerance = 0.000001;
     

	for (int i = 0; i < roots.Length; i++)
	{
	    float2 difference = z - roots[i];
     

		// If the current iteration is close enough to a root, color the pixel.
		if (abs(difference.x) < tolerance && abs (difference.y) < tolerance)
		{
			return colors[i]; // Return the color corresponding to the root
		}
	}
     

   }
     

   return black; // If no solution is found
     

• Julia set
• Mandelbrot set

• 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
• Newton's Fractal (which Newton knew nothing about) by 3Blue1Brown, along with an interactive demonstration of the fractal on his website, and the source code for the demonstration