(Question) Mandelbrot/Newton - Has it already been done?

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Offline mrrudewords

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« on: February 08, 2019, 05:52:44 PM »
I guess the answer is probably yes but I was messing about and combined the Mandelbrot and Newton formulas together and got something nice. I have added the formula along with many variations into Infinity Fractal.

EG:
Code: [Select]
P = power

Z = Z ^ P + C
Z = Z - (Z ^ P - 1) / (Z ^ (P - 1)) * P) * Relax


Linkback: https://fractalforums.org/share-a-fractal/22/mandelbrotnewton-has-it-already-been-done/2608/
Z = Z2 + C (obvs)

Offline claude

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« Reply #1 on: February 08, 2019, 07:06:09 PM »
Newton with a + C is called Nova http://www.hpdz.net/TechInfo/Convergent.htm#Nova

I tried to do a Nova of Burning Ship using Jacobian matrices for the derivative division, but it didn't look very good.  Much better was a simpler "Burning Nova" by:

Z → |Re(Z)| + |Im(Z)|
Z → Z - R (Z^3 - 1) / 3Z^2 + C

Changing the R value (complex) can give nice morphing animations...

My formula compiler for perturbation 'et' (which I use offline for parts of 'kf' too) can technically do Nova including the division, but the common subexpression elimination needs a lot more work to be performant enough to test it properly, and it takes ages to compile...

Offline mrrudewords

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« Reply #2 on: February 09, 2019, 09:35:17 PM »
Thanks for the info Claude.

Newton with a + C is called Nova http://www.hpdz.net/TechInfo/Convergent.htm#Nova

This isn't Nova though. Or is it some variant? It's not convergent as it uses the normal mandelbrot bailout check.

I tried to do a Nova of Burning Ship using Jacobian matrices for the derivative division, but it didn't look very good.  Much better was a simpler "Burning Nova" by:

Z → |Re(Z)| + |Im(Z)|
Z → Z - R (Z^3 - 1) / 3Z^2 + C

Burning Nova looks nice. I've put this in along with other variations, including Nova/Newton.

Changing the R value (complex) can give nice morphing animations...

One of the things lacking from my program at the moment is the ability to edit this value, along with other parameters.

My formula compiler for perturbation 'et' (which I use offline for parts of 'kf' too) can technically do Nova including the division, but the common subexpression elimination needs a lot more work to be performant enough to test it properly, and it takes ages to compile...

Perturbation, for now, is beyond me. I will try to look into it some time. I am always disappointed that I can't zoom very far before all the blockiness.



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