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) ∈ ℂ[Z]. When there are not attractive cycles, it divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, k = 1, ..., 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.

Each point of the complex plane is associated with one of the deg(p) roots of the polynomial in the following way: the point is used as starting value z0 for Newton's iteration z_{n+1}:=z_n-\frac{p(z)}{p'(z)}, yielding a sequence of points z1, z2, .... If the sequence converges to the root ζk, then z0 was an element of the region Gk.

To plot interesting pictures, one may first choose a specified number d of complex points (ζ1,...,ζd) and compute the coëfficients (p1,...,pd) 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 zmn = z00 + mΔx + inΔy, m = 0, ..., M - 1, n = 0, ..., N - 1 of points in ℂ, one finds the index k(m,n) of the corresponding root ζk(m,n) and uses this to fill an M×N raster grid by assigning to each point (m,n) a colour fk(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 |zD - ζk(m,n)| < ε for some previously fixed small ε > 0.

Newton fractal for three degree-3 roots (p(z) = z3 − 1), coloured by number of iterations required

Newton fractal for three degree-3 roots (p(z) = z3 − 1), coloured by root reached

Newton fractal for x8 + 15x4 − 16

Newton fractal for p(z) = z5 − 1, coloured by root reached, shaded by number of iterations required

Newton fractal for p(z) = z5 − 3iz3 − (5 + 2i)z2 + 3z + 1, coloured by root reached, shaded by number of iterations required

Newton fractal for p(z) = sin(x), coloured by root reached, shaded by number of iterations required

[edit] See also

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  • J. H. Hubbard, D. Schleicher, S. Sutherland: How to Find All Roots of Complex Polynomials by Newton's Method Preprint (2000), Inventiones Mathematicae vol. 146 (2001) – with a discussion of the global structure of Newton fractals
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