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I'm releasing my Max OS X Mandelbrot set program as freeware. Its not a fully featured product, its more of a proof of principle for using a kind of perturbation theory to speed up arbitrary precision fractal calculations. It should be good for zooms to 1e300 and further.

http://mac.softpedia.com/get/Math-Scientific/SuperFractalThing.shtml (http://mac.softpedia.com/get/Math-Scientific/SuperFractalThing.shtml)

Main Features:
Arbitrary precision Mandelbrot set images rendered in seconds.
Create deep zoom movies. (http://www.fractalforums.com/movies-showcase-%28rate-my-movie%29/mandelbrot-set-zoom/ (http://www.fractalforums.com/movies-showcase-%28rate-my-movie%29/mandelbrot-set-zoom/))
Create poster size images.

Requires Snow Leopard.

Hope you like it.

I'm no mathmatician and the wikipedia page on pertubation theory looks daunting.
Can you tell us a little about pertubation theory?
I infer it lets you calculate deep zooms with less iterations?

It works by having a reference point that is iterated in the standard way, using arbitrary precision maths. Once this is done, all the other points can be calculated relative to the reference point using only hardware floating point calculations, which is faster than the arbitrary precision maths. Also the initial iterations, which are usually well behaved and not chaotic, can be approximated with a power series, allowing SuperFractalThing to skip over them very quickly.

Hmm...
I was trying to work out a similar idea for deep zooms.
When you get up to doing 256 bit calculations what you're really doing at the processor level is lots of 32 bit calcs added together like long multiplication.
But the area of the image will be completely within the least significant 32 bit word, ie even 24000 (pixels) is a number smaller than 2^32.
So it would make sense to save every calculation result of every iteration step because all that really needed to be recalculated was the least significant word.
Of course the least significant still needs to multiply by the most significant through to the middle significant so your saving still having to do maybe 1/4 of the calculations.
Then theres the effect of carries possibly altering the 2nd least significant, numbers like .999999 that might trigger a cascade of carries and it all became too much
for my mental scratchpad.
Power series, okay thats getting a little more familiar to my high school maths  :educated:, at least I remember what the sigma summation symbol does, & I remember the teacher saying the word polynomial, but that was 30 years ago now :).

I've added a better download link. See opening post.

Cool, this thread is three years old.