Newton-Hines fractals

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Offline gerrit

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« on: April 06, 2019, 05:36:52 AM »
In "On Approximating the Roots of an Equation by Iteration" by Jerome Hines, Mathematics Magazine, Vol. 24, No. 3 (Jan. - Feb., 1951), pp. 123-127, a generalization to the usual Newton method for root finding is proposed. Actually several, but only one is suited to make into a classical Newton-type fractal.

For a given function \( f(z) \) the Newton iteration to solve \( f(z)=0 \) is \( z \leftarrow z-f(z)/f'(z) \), starting from some guess for \( z \). The Hines generalization involves a parameter \( k \) and reads  \( z \leftarrow z-\frac{f(z)}{f'(z)-kf(z)/z} \).

Jerome Hines was an interesting character. He was mostly a professional opera singer, one of the most famous ones during the second half of the 20th century. He was also a mathematician and besides the paper cited he also believed Cantor's theory of transfinite numbers and concepts such as uncountable infinities to be wrong. If that is not enough, he was also a born again Christian.

Anyways, I tried some values of \( k \) for the simple \( f(z)=z^3-1 \). The pixel encodes the starting z and the color is related to how many iterations before convergence (brighter means more) and to which of the three cube roots you converge from a given point. If no convergence is obtained color is black.

In the picture values of \( k \) are (in normal reading order) -0.47, -0.4, 0, 1, 2, 2.3, 2.44, 2.5, 2.6. Some of the images have been zoomed out a bit to show the whole pattern. There seem to be several discontinuities when \( k \) is varied.

An isolated value seems \( k=-1 \), with some totally random looking areas, see second image.

Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/newton-hines-fractals/2737/

Offline superheal

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« Reply #1 on: April 06, 2019, 06:34:55 PM »
I beleive that with k=1 you get Halley's method and with k=2 Schroder's method.
it is a nice find, and I will definately add it to my collection!
https://en.wikibooks.org/wiki/Fractals/fractalzoomer#Root_Finding_Methods

Offline superheal

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« Reply #2 on: April 06, 2019, 06:52:27 PM »
Here is an example for k = 0.15989847715736039+1.3020304568527918i

Offline superheal

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« Reply #3 on: April 06, 2019, 06:56:38 PM »
and here is a domain coloring render for 3 iterations,

Offline gerrit

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« Reply #4 on: April 07, 2019, 12:25:56 AM »
\( k=2.38 + 2.29 i. \)

Offline gerrit

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« Reply #5 on: April 07, 2019, 01:18:10 AM »
\( z^3-3z-1=0 \) with \( k = 2.45238 - 0.14286i. \)

Offline gerrit

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« Reply #6 on: April 07, 2019, 09:01:41 AM »
A degree 5 polynomial.

Offline gerrit

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« Reply #7 on: April 08, 2019, 03:11:33 AM »
A degree 200 polynomial with zeros at regular angles but random distance ([0 1]) from origin, and k=207.

Offline Dinkydau

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« Reply #8 on: April 08, 2019, 03:45:23 AM »
The last image is amazing.

Offline Fraktalist

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« Reply #9 on: April 08, 2019, 10:10:20 AM »
The last image is amazing.

agreed! this is awesome!

Offline superheal

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« Reply #10 on: April 08, 2019, 09:38:16 PM »
Familiar structures!
Nova Newton-Hines

with relaxation = 1
k = 1.0-1.0i
c = -0.6624365482233503+0.548223350253807i
Center: -1.3838386342005076+0.8029834727950508i 
Size: 0.01171875

Offline gerrit

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« Reply #11 on: April 09, 2019, 08:04:39 AM »
Two with 7th degree polynomial (roots at the 7 colored suns) and largish k.

Offline hobold

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« Reply #12 on: April 09, 2019, 11:07:43 AM »
The cool thing here, IMHO, is the fine line between strict deterministic math on one hand, and freedom of design on the other hand, when picking parameters of the root finding algorithm, and when picking the polynomial to solve.

Now I wonder if there would be a point in revisiting all the old bad root finding methods that have been superseded by newer and better ones. For fractal purposes, the old ones might suddenly be not all that bad.

Offline lkmitch

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« Reply #13 on: April 09, 2019, 06:33:02 PM »
Now I wonder if there would be a point in revisiting all the old bad root finding methods that have been superseded by newer and better ones. For fractal purposes, the old ones might suddenly be not all that bad.

For fractal purposes, actually finding a root is kinda boring. I think it's more exciting to look at the chaos that ensues when the method doesn't find a root. For example, take the standard Newton's method and add a complex factor k:

z = z - k f(z)/f'(z)

The standard method has k = 1; making |k-1| > 1 leads to some interesting images.

Offline gerrit

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« Reply #14 on: April 10, 2019, 05:04:36 AM »
The cool thing here, IMHO, is the fine line between strict deterministic math on one hand, and freedom of design on the other hand, when picking parameters of the root finding algorithm, and when picking the polynomial to solve.

Now I wonder if there would be a point in revisiting all the old bad root finding methods that have been superseded by newer and better ones. For fractal purposes, the old ones might suddenly be not all that bad.
There is the "simple iteration" \( z \leftarrow f(z)+z \); obviously it solves \( f(z)=0 \) when it converges, but it's too simple to be interesting I think.
I never got anything interesting out of the secant method which needs two initial values, but perhaps do a normal Newton step first iteration. Damped Newton has already been mentioned.


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