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

Given a function ** f[x]**, define
its

`g[x_] := x - f[x]/f'[x]`

If ** f[x]** is a polynomial, then

If x is a root of** f**, then

`g'[x] = f[x]*f''[x]/(f'[x])^2.`

It follows that if a root ** x** is
not a critical point (meaning that

The dynamics of the function ** g[x] **are
easy to analyze when

`f[x_] := (x - r1)(x - r2).`

Then g[x] has superstable fixed points at the
roots ** x = r1** and

**Newton fractals for cubic polynomials.**

The situation can be much more complicated when
** f** is a cubic. Suppose we have

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

- Explain why almost all points on the real axis are colored blue, meaning that their orbits converge to 1. Find the real values in the picture for which this does not happen.
- Explain why this picture shows large scale self-similarity with a scaling factor of 1.5.
- Fractint allows one to make similar pictures
for
for other exponents`f[x_] := x^d-1`. What is the corresponding scaling factor?`d`

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

** g[x_]** :

We see that ** g[1/2] = -1**, even
though -1 is the furthest root from 1/2, and

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

** g[Sqrt[1/5]] = - Sqrt[1/5] **and

`g[-Sqrt[1/5]] = Sqrt[1/5],`

so the orbit of ** Sqrt[1/5] **as period
2. Hence this point is fixed by the iterate

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.

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

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.

`CubicNewtonJulia2{; Douglas C. Ravenel
January 26, 1995 `

` ;
Use floating point option for this program. `

` ;
This is similar to the Newton-Julia set shown in the `

` ;
Mandelbrot-Lorenz video `

`z = (10^p2)*pixel, c = p1: `

`fz = z*z*z + (c-1)*z -c, `

`f1z= 3*z*z + c - 1, `

`z = z - fz/f1z, `

`|fz| > .0000000001 }`

Here we are iterating the Newton function for
** f[z_] := z^3 + (c-1)z - c**, and the iteration stops when

Figure 13.2. ** CubicNewtonJulia2
**for

- Does this image have large scale self-similarity, and if so why and with what scaling factor?
- Is there small scale self-similarity near
or other such points?`z = 1/Sqrt[5]`

By varying ** c**, we can get an analog
of the Mandelbrot set. The Newton function for

`g[z_] := (c + 2*z^3)/(-1 + c + 3*z^2).`

When ** c** is not 1,

`CubicNewton2{; Douglas C. Ravenel January
26, 1995 `

` ;
Use floating point option for this program. `

` ;
This is similar to the Newton-Mandelbrot set shown in the `

` ;
Mandelbrot-Lorenz video `

`c = (10^p1)*pixel, z = 0: `

`fz = z*z*z + (c-1)*z - c, `

`f1z= 3*z*z + c - 1, `

`z = z - fz/f1z, `

`|fz| > .0000000001 } `

As before we iterate the Newton function, this
time on the critical point ** z = 0**, until we get sufficiently
close to a root of

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

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.

- Find some miniature Mandelbrot sets on the real
axis in this picture. Plot the cubic curve for the corresponding value
of
and see what happens with Newton's method for`c`.`z = 0` - Find the periods associated with some miniature Mandelbrot sets in this picture.
- Try to find an equation for the main cardioid on such a miniature Mandelbrot set.
- Study the Newton-Julia sets corresponding to some points in some of these miniature Mandelbrot sets. see if there might be an analog of Lei's theorem (discussed in Lecture 8) for them.

**Newton fractals for some quartics.**

Suppose we have a quartic of the form ** f[z_]
:= z^4 + 6*a^2*z^2 + b**. Then

examines the family`QuarticNewton`for varying`f[z_] := (z^2 - 1)(z^2 + c)`. The image is concentrated near the positive real axis. It appears to be a mixtures of miniature`c`s and diamond shaped regions.`M`

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

examines the family`Quartic2Newton`, for which`f[z_] := f(z) = (z^2 - c)*(z^2 - 6 + c)`for varying`f''(z) = 12*(z^2 - 1),`. This image display large scale self-similarity and 6-fold rotational symmetry centered at`c`. The only solid regions appear to be minature`c = 3`s.`M`

Figure 13.5. ** QuarticNewton**
zoomed out by a factor of 20 with unmagnified detail at

Now here are some study questions.

- Why are these two images so different?
- Do the diamond shaped regions in Figure 13.5 have periodic orbits associated with them?
- Find the periods of the largest miniature
s in both pictures.`M` - Look for preperiodic points in both pictures.
- Look at Julia sets (using
and`QuarticNewtonJulia`) associated with such points.`Quartic2NewtJulia` - Look for a main body equation in both pictures.
- Look for other interesting 2-dimensional slices of this 4-dimensional Newton-Mandelbrot set.

*This page was last revised on March 23, 1998.*