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In these notes I will use mathematical notation similar to the syntax of Mathematica.

13. Fractals associated with Newton's method.

We have discussed Newton's method before. It is a mthod for finding the roots of polynomials or other functions. Given a function f[x] with a root near some x0, we can get a better approximation to the root by replacing the estimate x0 by x1 = x0 - f[x0]/f'[x0]. By repeating this procedure a few times we can usually get a a sufficiently precise estimate of the root in question, but not always.

Given a function f[x], define its associated Newton function to be

If f[x] is a polynomial, then g[x_] is a rational function, i.e. a fraction in which the numerator and denominator are polynomials. It is not defined when f'[x] = 0, i.e. at critical points of f. Applying Newton's method for the function f[x] is the same thing as iterating the function Nf[x].

If x is a root of f, then g[x] = x, i.e., x is a fixed point of g. The derivative of g is

It follows that if a root x is not a critical point (meaning that f'[x] is not zero) then it is a superstable fixed point of g. It also follows that if x is a point of inflection of f (meaning that f''[x]=0) and not a critical point of f, then it is a critical point of g.

The dynamics of the function g[x] are easy to analyze when f is the quadratic function

Then g[x] has superstable fixed points at the roots x = r1 and x = r2. The orbit of any point that is not equidistant from from the two roots will converge to the nearer one.

Newton fractals for cubic polynomials.

The situation can be much more complicated when f is a cubic. Suppose we have f[x_] := x^3 -1, so the roots are 1, (-1 + I*Sqrt[3])/2, and (-1 -I*Sqrt[3])/2. The following picture, hioch was produced by Fractint using newtbasin, colors each pixel according to which of the three roots its orbit converges to. Each colored region is called a basin of attraction for the root in question. Note that wherever two colors meet, the third color can be found in between them.

      

Figure 13.1. Newton basins of attraction for the function f[z] = z^3 - 1 with magninfication .666 and .05.

Here is another example. Suppose we have f[x_] := x^3 - x, so the roots are 0, 1 and -1. Then the Newton function is

We see that g[1/2] = -1, even though -1 is the furthest root from 1/2, and g[-0.45541] = .5, so g[g[-0.45541]] = -1 even though the nearest root is 0. Thus the orbit of x need not converge to the nearest root.

What is worse, the orbit might not converge to any root. For example

so the orbit of Sqrt[1/5] as period 2. Hence this point is fixed by the iterate g2[x_] := g[g[x]]. Is it an attracting fixed point?

What can we say about the Julia set of g? There are two things about it that make the question quite different from that of the Julia set associated with polynomial function.

  1. The function g[x] is undefined when 3x^2 - 1 = 0, i.e., when x = 1/Sqrt[3] or x = -1/Sqrt[3]. This means that if any iterate of g sends x to one of these two numbers, then the orbit of x is undefined.
  2. Since Limit[g[x]/x, x -> Infinity] = 2/3, if x is large, then g[x] is approximately 2x/3, which is smaller than x. Thus there are no unbounded orbits.

Thus the Julia set as defined previously consists of all points except those with undefined orbits.

A more interesting question is which points have orbits that do not converge to a root of f, i.e. for which points does Newton's method fail to produce a root? For this we use the following Fractint code, taken from dcr95.frm.

Here we are iterating the Newton function for f[z_] := z^3 + (c-1)z - c, and the iteration stops when f[z] gets sufficiently small, i.e., when the orbit gets close to a root. The parameter p2 is included to allow arbitrary zoom-outs, which are not possible otherwise.

  

Figure 13.2. CubicNewtonJulia2 for c = 0, with 10-fold magnification near z = 1/Sqrt[5].

By varying c, we can get an analog of the Mandelbrot set. The Newton function for z^3 + (c-1)z - c is

When c is not 1, g has a critical point at z =0 in addition to ones at the roots of f. The orbits of the roots are obvious; they are fixed by g. The other critical orbit, the one for z = 0, may or may not converge to a root. The following Fractint code (in dcr95.frm) tests this.

As before we iterate the Newton function, this time on the critical point z = 0, until we get sufficiently close to a root of f. The magnification parameter is p1.

  

Figure 13.3. The Newton-Mandelbrot set for f[z] = z^3 + (c-1)z - c with detail near c = .15 + 1.7*I.

The surprising thing about this image is the appearance of copies of the classical Mandelbrot set within it. This was first reported by Curry-Garnett-Sullivan in 1983. Here are some possible topics for a group project.

Newton fractals for some quartics.

Suppose we have a quartic of the form f[z_] := z^4 + 6*a^2*z^2 + b. Then f''[z] = 12(z^2 - a^2), so there are points of inflection at z = a, and z = -a. Hence these are the two orbits that have to be tested to construct the Newton-Mandelbrot set. However, since f is an even function, these two orbits behave in the same way and only one of them has to tested. This is convnenient because Fractint seems to have trouble when testing two orbits. As in the case of cubic polynomials, this family of functions is apramtrized by two complex variable a and b, so it leads to an unviewable 4-dimensional fractal. We can only look at 2-dimensional slices of it. Here are some examples in dcr95.frm.

  

Figure 13.4. QuarticNewton zoomed out by a factor of 10 and magnified 30 times at c = 7.96.

  

Figure 13.5. QuarticNewton zoomed out by a factor of 20 with unmagnified detail at c = -1.25.

Now here are some study questions.


 

This page was last revised on March 23, 1998.