Concerning Newton Raphson Zooming...

[stardust4ever]

First time using this feature. I'm currently digging for a minibrot. I was a bit past 2^1200 and the iteration depth already around 14 millions, reference orbits were taking painfully long for frame by frame zooming.

So I opened up Kalles Fraktaler and loaded the coordinate to try out Newton Raphson zoom technique. I was already about 20 zooms past the target formation that I was trying to double (to make sure it doesn't select the wrong centroid), so I loaded 3/4 setting to double the figure. After some time, KF discovered the period to be 11914564! So it started pass 0, then pass 1, now it is finished with pass 1 and now it is 22% into pass 2. How many passes does it take to locate?

Also any possibility this could get multithreaded? I have no idea how it is done but probably going to go to bed soon...

[Kalles Fraktaler]

It usually finds the minibrot in less than 10 Newton-raphson iterations/passes.
The time grows exponentially. Because of higher periods, and also higher precision.
So period of 11 million together with e1000 probably takes several hours or more than a day.
The only consolation is that it's faster than manual zooming... 

[stardust4ever]

I corrected my post. I'm actually around 2^1200 or so. I said e1000 but I meant zooms. Why are multiple passes needed? Sorry I'm a bit ignorant on how it works. If my predictions are accurate, there will be a location somewhere north of 3000 zooms (e1000 ballpark) at approximately 85 million iterations. Anyway I am using the new minibrot locatoin technique to replicate the image rather than manual zooming. Work smarter, not harder, as they say. Mandel Machine may struggle a bit to render this location to extreme resolution despite my having 16 gigabytes of RAM to work with. I am attempting to carve a "Magnum Opus Ex" style formation out of this:
http://www.fractalforums.com/ultra-deep-mandelbrot/dance-of-the-fire-dragon/

Every layer of columns in the image increases the bailout by about 1570 iterations. Wish me luck...

BTW, here's a pretty good discussion on locating minibrots with a given period. Apparently the exact location of the minis for a given period are solutions to complex polynomial equations.
http://math.stackexchange.com/questions/404066/help-locating-mini-mandelbrots

For period 3, is not so bad: f_c(0) = (C^2 +C)^2 +C
However with each period increase, an extra (...)^2+C is placed around the equation. I imagine the solution can get really nasty with 14 million nested pairs of parenthesis.

[Adam Majewski]

http://www.fractalforums.com/theory/the-mandelbrot-polynomial-roots-challenge/

[Kalles Fraktaler]

Why are multiple passes needed?

Because Newton-Raphson is an iterative numeric solving method.
For each lap, check how close the function results to zero.
When good enough, break.
I should be able to split the iteration in two threads, one for f and one for f'.
Perhaps it is possible to tweak the precision needed and the good enough value.
But not much in afraid

[stardust4ever]

I'd like to report a bug in the algorithm. I opted to double the pattern (3/4 to the minibrot) from the original formation located at 1219 zoom levels. First I zoomed about 20 or so zoom levels into the point where I wanted to zoom into, to insure the correct path was chosen. After 5 passes and some hours later, it honed in on a position at 1519 zoom levels. That's only a quarter of the path to the minibrot, not half! It should have placed me at the target formation around 1800 or so.

So I am redoing it. Lesson learned, this time I will tell it to locate the minibrot, then manually back out to a distance halfway between the minibrot and the original location. Still it is useful even if unoptomized and non-multitreaded, because I don't have to baby it all the way to the target formation. Even Mandel Machine is taking over a minute just to calculate the reference and Series approximation.

Edit: Kalles fractaler found the minibrot. I awoke to find my PC burning with KF at 3% on a black image some e600+ deep, 225 million iteration depth. I saved the file and copied the coordinates into notepad for use with Mandel Machine. The minibrot is not always located exactly twice the zoom depth of the forked path, and can occasionally be at a shallower depth if the Julia formation zoomed into is sufficiently complex. I can take the depth of coordinates using log function on calculator to determine the exact formation at equidistant zoom depth to the fork point and minibrot, expressed as log base 2 or "zooms". Load coordinates in Mandel Machine to frame and render the object. This is certainly more convenient to manual advance as I am not tethered to the PC for hours on end.

EDIT2: Well working out the exact location was easier than I expected. I saved the KF file and loaded it in Mandel Machine. The image was solid black with bailout of 225 million. I took the minibrot location (2027 zooms) and the fork location (1219 zooms) and calculated the point exactly halfway between those two (1623 zooms). This placed me smack dab in the center of the target formation!


As it turns out, the 1519 figure was "wrong" because I forked into a highly detailed Julia formation. The Minibrot (2027) was much closer than the expected 2438, because of this fact. So 3/4 to the minibrot is not always accurate. The period duplicity is always equidistant to the fork formation and the final minibrot.

I should come up with something to show a bit later. Peace...