Newton fractal

The Newton fractal to a non- constant meromorphic function which maps the complex numbers in itself, is a subset of the set of complex numbers. More specifically, it is the Julia set for the function

The Newton's method for finding zeros of the function describes. Newton's method itself is constructed from a starting value of a sequence of recursion.

Depending on the starting value, the orbit of

Show quite different behavior.

Note: Here the exponent refers to as a function, and not to its value. therefore, means the -fold iterated application of formally so.

You simply select a second complex number in an environment of and compares their orbits. Then there are exactly two ways in which the sequence

Can behave:

In the first case lies in the Fatou set. In the second case lies in the Julia set. Since Julia sets are fractals down to some excellent exceptions, and is derived from the Newton method for, also called Newton Fractal of.

Importance for the Newton's method

If the start value of the Newton iteration near a simple zero of, then the method converges quadratically to this root. At a multiple root, the Newton method is nevertheless still linearly convergent. These cases are part of the case 1 from above.

The starting value, however, is closer to the Newton fractal, the unmanageable is the result of Newton's method:

  • Even starting values ​​that are far from a zero point can converge toward the latter, even if other zeros lie much closer to the initial value ( Case 1 ).
  • There are initial values ​​that do not converge to a zero, but only against a periodic cycle ( Case 1 ). An example of the polynomial. There are initial values ​​, which are captured by attracting cycle { 0,1}.
  • If the start value in Newton Fractal himself, then he does not converge to a zero ( for case 2).

For non-constant meromorphic functions, the Newton fractal is a null set, that is, it is a set of measure zero. So Newton's method leads for almost all initial values ​​to an attracting periodic cycle of f out. If this cycle has period length 1, then it involves a zero of. Otherwise, the Newton's method is not convergent this starting value. Although the Newton fractal is a null set, so it can give all the territories in which the method does not converge to a zero.

This applies also for real-valued rational functions. In turn, the polynomial is used as an example. Because it has real coefficients, the values ​​of the Newton iteration for real starting value remain real-valued. Since the real axis passing through regions of the non-convergence, there are intervals for which there is no convergence. From such intervals there are infinitely many.

Beispielfraktale

Example 1

The figure at right shows the Newton fractal ( in white) to color-coded according to speed of convergence and the three zeros. Starting values ​​that are in the beige -drawn areas converge to the same zero point (on the left in light beige), similarly for the green and the blue area. The zeros of the green and blue region are symmetrical to the horizontal symmetry axis of the right. Converges faster a starting value to its zero point, the brighter it is dyed. The values ​​in the infinitely many red areas do not converge to a zero, but are captured by the rising cycle. The Newton fractal - visible in the image as bright structure - is not limited. In the three directions to be recognized, it reaches up to ∞.

Example 2

The figure at right shows the Newton fractal to a polynomial with 7 random zeros (white dots ), the range represents the fractal itself is eg the edge of the yellow area. Also it is the edge of the green area, the edge of the turquoise area, etc. This property is all Julia sets in common. The colors red and pink were used twice, so that the boundary of the red and pink territory does not correspond to the Newton fractal.

Attractor Inventiones Mathematicae
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