*Continued* SuperFractalThing: Arbitrary precision mandelbrot set rendering in Java.

[stardust4ever]

I did it automatically since the minibrot was already dominant (e.g. the period is the same for all pixels) so I knew which polynom's zero crossing to compute. Takes some time because the Newton steps must be done at full arbitrary precision.

Is there a software app I can use for the 'brot finder? It would save some time for sure.

[cKleinhuis]

people, i would like to split this topic and lock it down, where would you place the cut ?!

[stardust4ever]

people, i would like to split this topic and lock it down, where would you place the cut ?!

Sorry bout that...

[cKleinhuis]

np, the thing is to keep the original topic intakt whilst enabling an ongoing discussion, overlong topics are of no use for people, since i knew the originating thread was like 2 years old i splittet it at the longest pause between posts, just continue as nothing happened, just some minimal effort cleaning up in the meantime

i just think the originating topic is one more super-epic topic here in the forums, beside of the famous mandelbulb/mandelbox topics where the development of such can be traced, and this pertubation technique - although long existant before we found it, and it has been used before by kais power tools fractal implementation - this thread marks the development of useable and available mega ultra deep zoom mandelbrots possible...

[Pauldelbrot]

I did it automatically since the minibrot was already dominant (e.g. the period is the same for all pixels) so I knew which polynom's zero crossing to compute. Takes some time because the Newton steps must be done at full arbitrary precision.

Who needs Newton steps? Once you know the dominant period p just find the pixel for which the pth iterate is of smallest magnitude. The minibrot should be inside that pixel somewhere. The software can auto-zoom to a few-pixels neighborhood of there, then repeat calculating the first p iterates of this new image to find the next zoom target point.

[stardust4ever]

In this location the central dominant minibrot has period 418864. Computing the zero crossing of the corresponding polynomial locates the minibrot at

-1.76979700322139811591272513043899832799423369499068746040312321369139476279899734327685384106424938431439273576680330733704966546075580838901324891220246239218903287505782319765936273238087369689487534737359516124840715760630396132975573610932201163074628687245503337178276171115248596381484098549511985811224780956321700144001233548139295889127740464191577029223476957057942352608361586911947339765514426923055404845140828712983972948274581253682130400984935617578642192675431716605409501767773747890962982410145941148467865154044608549657935615408744476886410714406890349574710784014258749496483079037310546638701763780494020009322694833109833656402410119130478284600925109395602405485985011438094250629579927270304012249169584818855490091011034850066008814214293599691799941578041340907231850565831837098638971449938935994601792205438960554930723986381877122351711795882803085844823543736994077850454865580941408628641027809410360282931245336574301206947989732268717006195367435719086670011251760720899568816751908549316856858712898480478800635959347100781293499250828473881321840106718612921692041981341359850708691437845116651465935653020129685931665064112991181637664436069589912197864687625835231334856460972500730321507970263314589963166350417424706366261835720179449175566433458116106325171826646992999680483823690344872849669066814331960087408951512529176426834553498117497629197785569880574692522939972961522510960524534583072265551760614774450799723561044615076588827984931672903629230164610169826241538784865555145381338917258229559017138074679046545750565703569290153270887791912366870238890702486377674493961627842425415072641536223340784982438486048756109238181153075391103742999718461989487988255182749425809658290851105686957800331487046619356847741786931568734133797812990312933679468689355633257241932332586807751783991361005487951858068862626827875513314445086552403572026135269341415265563914895613317095945080129111249617399947471951570374601941026322586575889678534000148411715548247602090863178460885536238487047026969052782268862081294620522011538188275677933094565746782844741895263212598E+00
4.503808149118977453591027370762118116191847489651632102771075493630536031121753213019458488948070234821894347490919752 32128719902266967792409275276218671134664739202538733880630147980377066457243173553858784184258065626405478713476529943 75685863015511904074453632654407731289619946868720085884280405841386804671414034982833768121999000401733388984737998508 35523341852444210373993799979274072458522457971439601401283190488219977380751679864657632594486990141780409069050808535 33679083210095437351400022620788443700681865056074859184889623921225508741770547501475133877301147491846294015630493195 94413147950329230917914373568299313895801070552430312839787385413077643433921434686758800882730741386718858427487804801 73527152642383437688144097648231731279522222357988455250353865370120443546331395472996006556618614941953429666058354649 10451202485512530423175907298924572677884684325102852936015719933302605823099958630951988450410491580664701963842251461 35190645341340161891884063141465638742680614101092435645795624718302058131414609501281021540435472453888874524109018147 02121578711328524425442226752168664749086242203613749999027884515745350840633982861734634138141253642303937961493945458 38176191438823739844915158113285022936463789829746280707055929391192625872076997627990447836359937976951672647199177818 72517689037585583899463944250055017306480718807197254236743510423432718914191161718864625412816080818679138546319519759 89748541205329675986737013154577653006827691952880225127567357459621316524513472420563020300861878311519895655738526548 29737784116356975937395880502857287215780402078167418834602295096014173047038182390355477059048628119343002217338189674 84428900612407421285966391654470156922336601567981570299684787648714514350236588685564191491795576963451396365624203611 89623693814216660262167258794137460777065623334381376669587093792227710384619914833779522355034279775231366236846879929 65077410226071259699613708732240144706025226046260403350230398624904436384826525344982234790191805054228954439652523570 302168603714064304338213704267058855190821114715226120756650420403035424042014899248446596447E-03
2.0E-2105
There is another juncture at 1.7E-1851 .
Now a question for Kalle and other fractal champions: How much can you skip near the minibrot?  
A program option suited best for your needs would allow you to click on a pixel where you want to use a minibrot deep down in the neighbourhood of that pixel, locate the minibrot, compute one or more fork locations in between and generate images of the fork locations for you so you can click again on other pixels to deviate from the central minibrot. And so on. Pretty straightforward to implement.

Your coordinates worked like a charm!

[hapf]

Who needs Newton steps? Once you know the dominant period p just find the pixel for which the pth iterate is of smallest magnitude. The minibrot should be inside that pixel somewhere. The software can auto-zoom to a few-pixels neighborhood of there, then repeat calculating the first p iterates of this new image to find the next zoom target point.

That's an interesting idea, but for the iterations to be fast and use perturbation one needs a reference. Where is the reference coming from? Ad hoc references need full precision iterations as well. And how long can it last when you zoom in repeatedly? If one uses a minibrot we are back to square one. Are you suggesting full precision iterations all the time? If not, what reference should be used?

[Pauldelbrot]

That's an interesting idea, but for the iterations to be fast and use perturbation one needs a reference. Where is the reference coming from? Ad hoc references need full precision iterations as well. And how long can it last when you zoom in repeatedly? If one uses a minibrot we are back to square one. Are you suggesting full precision iterations all the time? If not, what reference should be used?

Compute center of image to full precision to p iterations. Compute rest with perturbation and find minimum of iteration p. Repeat.

[claude]

Repeatedly rendering zoomed in images to find the minibrot will take linear time, because you get only a constant few more bits of precision each time.  Newton's method has quadratic convergence when you get close enough, which means the number of accurate bits doubles each time.  One iteration of Newton's method will take around twice the time of calculating the center of the image to p iterations at high precision, let alone calculating the image pixels.  So I think Newton's method will be a lot more efficient.

[Pauldelbrot]

Maybe. It depends how much high precision arithmetic actually is needed, how fast convergence actually is under these circumstances, and the like. I have had very odd and unreliable results trying to use Newton's method for any polynomial of degree much above 45. That would limit its utility to minibrots of low period.

[hapf]

Maybe. It depends how much high precision arithmetic actually is needed, how fast convergence actually is under these circumstances, and the like. I have had very odd and unreliable results trying to use Newton's method for any polynomial of degree much above 45. That would limit its utility to minibrots of low period.

I use a modified version of Newton regularly for periods in the 10000s and more. Even 100000s work (see above example). Of course large periods can take minutes and more to finish, but it works quite well. From a point one can usually find several minibrots with a period < point's escape iteration number, including the dominant period. Can be less than five, can be dozens, depending on the location. Some periods will quickly or (grumble!) later on fail (due to values going to infinity), others can loop forever, still others converge after many Newton iterations. But usually it takes less than 20 iterations, often less than 10. Adequate bits are necessary or the search fails. Have not had a location yet where no minibrots in the 'hood could be found. One can also go minibrot hunting from a point with a target period >> point's escape iteration by using Newton iteratively and switching horses on the fly as needed, so to speak. It's fun clicking near the tip at -2 and ask for a period 34000 minibrot. 

[stardust4ever]

I use a modified version of Newton regularly for periods in the 10000s and more. Even 100000s work (see above example). Of course large periods can take minutes and more to finish, but it works quite well. From a point one can usually find several minibrots with a period < point's escape iteration number, including the dominant period. Can be less than five, can be dozens, depending on the location. Some periods will quickly or (grumble!) later on fail (due to values going to infinity), others can loop forever, still others converge after many Newton iterations. But usually it takes less than 20 iterations, often less than 10. Adequate bits are necessary or the search fails. Have not had a location yet where no minibrots in the 'hood could be found. One can also go minibrot hunting from a point with a target period >> point's escape iteration by using Newton iteratively and switching horses on the fly as needed, so to speak. It's fun clicking near the tip at -2 and ask for a period 34000 minibrot. 

@hapf Glad you've created a solution for finding minis. It seems like I could really use a software algorithm for locating minibrots. Would save a ton of time. Do you have something available with a decent GUI I could use?

[hapf]

@hapf Glad you've created a solution for finding minis. It seems like I could really use a software algorithm for locating minibrots. Would save a ton of time. Do you have something available with a decent GUI I could use?

I'm afraid not. I'm not even on Windows. This functionality would be best added to Kalle's or Botond's program so it's integrated with the rest.

[stardust4ever]

I'm afraid not. I'm not even on Windows. This functionality would be best added to Kalle's or Botond's program so it's integrated with the rest.

If this thing could be added sometime, that would be great. Thanks.

[Kalles Fraktaler]

this pertubation technique - although long existant before we found it, and it has been used before by kais power tools fractal implementation

I've read this before but I haven't seen any references to it. Is there any?
If not I am sceptic. And did it use series approximation? And without pauldelbrot's glitch solving method it wouldn't be much useable anyway...