### Perturbation theory

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• 3f
• Posts: 1545

#### Re: Perturbation theory

« Reply #15 on: November 05, 2017, 05:09:17 AM »
[quote author = gerrit] Isn't there a +ϵ0 missing in the second and 3d formulas in that (Pauldelbrots discovery post) post?[/quote]
Hm, I think you're right.  Well spotted, perturbing d_0 should give d_0+e_0 indeed!
Thinking again, it does not matter, for when $$\delta_n \approx -z_n$$ while $$z_n$$ is not small,
$$\delta_n$$ must be so large that even the $$\delta_0$$ term will drop out of the floating point calculation, let alone the missing $$\epsilon_0$$.

So the "snapping together" argument is valid, in a "hand waving" sense, as there is no $$\epsilon_0$$ to push the orbits apart. What's missing is how to relate this to FP errors, in order to justify the choice of the tolerance. I've been playing with it and it works brilliantly. I assume if any glitch was discovered that does not get caught by this method I would have found it?

I found your old post relating it to loss of bits, but am not sure how bit loss in the $$z_n+\delta_n$$ addition would be a problem. I like the "influence" argument of hapf I dug up from the old forum (lost the link): regions under the influence of the reference mini will follow its orbit until they bail out, so if the Pauldelbot criterium is triggered that is a signal this point is under the influence of a different period mini and needs a different reference orbit. It's amazing that things don't fall apart before the orbit tells you that it does.

• 3d
• Posts: 977

#### Re: Perturbation theory

« Reply #16 on: November 06, 2017, 03:59:19 AM »
I found your old post relating it to loss of bits, but am not sure how bit loss in the $$z_n+\delta_n$$ addition would be a problem.

I was probably confused and not properly understanding Pauldelbrot's "snapping together" argument at that time...

• 3f
• Posts: 1545

#### Re: Perturbation theory

« Reply #17 on: November 06, 2017, 05:22:26 AM »
I came up with a better way (I think) to find the first reference point.

Instead of using the 4 corners of your image and iterate till they surround the origin, add one more point, the center.
Then while iterating the 5 points which divide the viewing rectangle into 4 nonoverlapping triangles find periods in all 4 triangles and select the one with highest period. You can do even better by continuing the period finding algorithm, after the lowest period is found, until one of the 5 points diverges; this will usually find some additional higher period minis, sometimes many. Select the highest period.

The example below has 157377 glitches at 1280X720 when using the rectangle method to find the reference point, 939 when using the 4-triangles method.

I found thus far that if, after correction of all glitches, you redo from scratch using the highest period mini used in glitch corrections, the whole image renders glitch free. Is that generally true, or are there complicated exceptions? I can't go to deep/complicated in MATLAB unfortunately.

#### hapf

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#### Re: Perturbation theory

« Reply #18 on: November 06, 2017, 09:32:17 AM »
Going for the highest period reference has been known for years to generate the least glitches. But it is not a solve all problems recipe. Depending on the region you can still end with many glitched areas. And if you go for the highest periods it's not very effective to go for the lowest periods first.

• 3d
• Posts: 977

#### Re: Perturbation theory

« Reply #19 on: November 06, 2017, 01:10:03 PM »
FWIW mandelbrot-pertubator uses 103 references for this location, but adding non-primary references is very cheap in this renderer.

The example below has 157377 glitches at 1280X720 when using the rectangle method to find the reference point, 939 when using the 4-triangles method.

Can you try this as initial reference? (I would myself but my current renderers are not set up to take primary reference location as input and it would be quite some work to retrofit this capability...)
Code: [Select]
-1.99999999913827011872293576313006855728014625181797324868265679733640041468660679970507390829192649343970240149934790854427123302445527558173337072695534623e+00-3.22680215517268694173676401004420591482010166625171033690538595372400588672285016792299922603720344996212603521835265519194915997936965950653063566007741461e-19This is (near, to 512bits of precision) the Misiurewicz point at the center of the spiral, with preperiod 111 and period 17.  As Misiurewicz points are repelling, you may need to use higher precision for the reference than you would for an attracting minibrot nucleus.  You can use Newton's method to get a higher precision location if needed, example code here: https://code.mathr.co.uk/mandelbrot-numerics/blob/HEAD:/c/lib/m_d_misiurewicz.c (the simpler "naive" method is fine if you have a good initial guess like the location I provided, you'll need to adapt it to provide a suitable epsilon2 based on the desired precision)
Here is one of its external angle,s on my mandelbrot-web app:  link
« Last Edit: November 06, 2017, 01:41:44 PM by claude »

#### hapf

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#### Re: Perturbation theory

« Reply #20 on: November 06, 2017, 03:37:18 PM »
The Misiurewicz point renders this without glitches. Interesting. While my program renders this without glitches and one minibrot reference  it will not last to any depth when the reference leaves the picture area. The same will apply to the Misiurewicz point no doubt. But which one lasts longer? I have to do some tests some time.

• 3f
• Posts: 1545

#### Re: Perturbation theory

« Reply #21 on: November 06, 2017, 04:32:23 PM »
Going for the highest period reference has been known for years to generate the least glitches. But it is not a solve all problems recipe. Depending on the region you can still end with many glitched areas. And if you go for the highest periods it's not very effective to go for the lowest periods first.
Thanks for the info.
I don't understand your last sentence. Do you mean there is a way to find higher periods without finding lower periods?

#### hapf

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#### Re: Perturbation theory

« Reply #22 on: November 06, 2017, 04:53:15 PM »
Thanks for the info.
I don't understand your last sentence. Do you mean there is a way to find higher periods without finding lower periods?
One can always try to find a minibrot from any pixel with a period a bit smaller than the escape iteration of that pixel.

• 3f
• Posts: 1545

#### Re: Perturbation theory

« Reply #23 on: November 06, 2017, 07:17:57 PM »
Can you try this as initial reference?
I got 5 glitches, which are probably false positives. You can zoom in almost indefinitely on the dust. Up to 1e8 additional zoom still give a few or no glitches.

If you zoom into the center cross, glitches appear around zoom of 100, and then things fall apart rapidly.

• 3f
• Posts: 1545

#### Re: Perturbation theory

« Reply #24 on: November 07, 2017, 12:34:14 AM »
Going for the highest period reference has been known for years to generate the least glitches.

Attached seems to be a counterexample to that proposition. Using the period 1602 mini in the center of the  largest island on the left gives 1088 glitches (threshold 1e-3), whereas using the period 3186 mini in the smaller island on the right gives 1602 glitches.

I guess that kills the idea of using 4 triangles and/or continuing the period finding.
« Last Edit: November 07, 2017, 01:01:27 AM by gerrit »

#### hapf

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#### Re: Perturbation theory

« Reply #25 on: November 07, 2017, 01:15:23 PM »
Attached seems to be a counterexample to that proposition. Using the period 1602 mini in the center of the  largest island on the left gives 1088 glitches (threshold 1e-3), whereas using the period 3186 mini in the smaller island on the right gives 1602 glitches.
The green pixels are glitches? Are they green because you coloured them so because they are glitches? Or because the regular colouring gave them this colour? How did you decide they are glitches? When I use the left reference I have no glitches at this resolution. And I mean numerical error and not triggering the Delbrot measure at 10e-3.
The highest period does not give generally the fewest glitches in a triggering the Delbrot measure at some value sense. Or even the fewest areas with real numerical error beyond some threshold. It also depends on other circumstances. Such as is there a dominant period in the whole image (here it is except some pixels) or is it mixed? In which neighbourhood are we going for the highest periods (for example in the circles around a minibrot each circle can be selected for a highest period but which circle should we use, if we don't use the central minibrot which has a lower period compared to minibrots in circles?)?
Finally, why do I see these kfr regions horizontally mirrored compared to the pics here? Do I have a sign bug somewhere?

• 3f
• Posts: 1545

#### Re: Perturbation theory

« Reply #26 on: November 07, 2017, 05:46:29 PM »
hapf: glitch measure in previous post was Pauldelbrot measure with tol=1e-3.
Do you have a better measure? "No numerical error" would make every pixel a glitch, unless you do everything in symbolic math.

• 3d
• Posts: 977

#### Re: Perturbation theory

« Reply #27 on: November 07, 2017, 06:54:00 PM »
Finally, why do I see these kfr regions horizontally mirrored compared to the pics here? Do I have a sign bug somewhere?

KF has the sign bug (it has imaginary increasing downwards instead of upwards).

#### hapf

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#### Re: Perturbation theory

« Reply #28 on: November 07, 2017, 08:04:35 PM »
hapf: glitch measure in previous post was Pauldelbrot measure with tol=1e-3.
Do you have a better measure? "No numerical error" would make every pixel a glitch, unless you do everything in symbolic math.
When I say no numerical error I mean the numerical error is below some small threshold like 0.1% and certainly irrelevant for how the fractal looks after colouring. I have no better measure for identifying potentially problematic pixels than using the Delbrot quotient(s). My point is that it is a heuristic and when it triggers at some value like 10e-3 it is neither necessary nor sufficient. So you can't use it to know how many glitches there really are. That requires visual inspection and when in doubt comparison with the positively correct value (as computed with arbitrary precision).

• 3f
• Posts: 1545

#### Re: Perturbation theory

« Reply #29 on: November 07, 2017, 11:33:25 PM »
Going for the highest period reference has been known for years to generate the least glitches.

Here's a clearer counterexample to that proposition. The first image is computed in full precision, then one using PT with a single reference mini of period 196 (indicated in plot by black square, 128 Pauldelbrot glitches), the last one using a period 277 mini as reference (15532 glitches).

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