Perturbation calculations

Ultra Fractal supports arbitrary precision arithmetic to zoom as deep as you want, without hitting a precision limit. The drawback is that the further you go, the slower the calculations become. As soon as you go past the regular precision level, images typically take hours or even days instead of minutes to complete.

For some fractal types (for example Mandelbrot), it is possible to dramatically speed up deep zooming by a new calculation algorithm based on perturbation theory. This algorithm was discovered by K.I. Martin and first implemented in SuperFractalThing. Ultra Fractal now features this perturbation algorithm with further advancements to make it more user-friendly.

You don't have to do anything to enable perturbation calculations: if the formula supports it, deep zooming just becomes orders of magnitude faster as if by magic.

How does this work?

Behind the scenes, the perturbation algorithm starts by calculating one pixel at the center of the image with full precision, storing all intermediate values for the main "Z" variable of the fractal formula. This is called the reference orbit. All other pixels are now calculated as a small deviation from the reference orbit ("perturbing" the reference orbit). The beauty of the algorithm is that this step can be done using regular double floating-point precision, which is fast.

Nothing is perfect though: sometimes the regular double precision will not be enough to follow the orbit of a neighboring pixel closely enough. In this case a new reference orbit at this point is needed, because the original reference orbit differs too much. Most programs implementing the perturbation algorithm require you to manually examine the image and discover that it's unlikely to be correct (it will usually contain large blank areas or "blobs" where the precision was too low). The problem with this is that you can never be fully certain that the image is correct.

The algorithm in Ultra Fractal has been extended to detect these precision problems and generate new reference orbits automatically. No manual intervention is needed and it's not even possible to manually add reference points. All that's left is worry-free deep zooming. However, you can adjust the precision tolerance for the algorithm: see the note below.

For more information regarding the underlying mathematics, see the SuperFractalThing web site.

Notes

You can check whether the perturbation algorithm is used in the Statistics tool window: the Precision reading will show a value like "Arbitrary (Perturbation: 99.7%)". This indicates that the perturbation algorithm is active and it was able to skip 99.7% of all regular arbitrary precision calculations (see How does this work? above).
If perturbation calculations are used, the Periodicity Checking setting in the Formula tab of the Layer Properties tool window is replaced by a Perturbation setting that sets the precision tolerance for the perturbation algorithm. Normally this should be left at the default Precise setting. For some images, this will result in a very slow calculation because too many different reference points are added. In this case, the efficiency percentage displayed in the Statistics tool window will show a low number like 60% instead of around 99%. You can then try the Fast or Very Fast options to see if this still generates a good result but with much greater speed.
Many formulas can be updated to support perturbation calculations. If you are a formula author, see Perturbation equations in the Writing Formulas chapter to find out how to implement perturbation support in your own formulas.
The SuperFractalThing paper also discusses a techique called series approximation to further speed up calculations. Ultra Fractal uses this in the Mandelbrot (Built-in) formula, but it is currently not available in other formulas. Series approximation is hard to generalize to other fractal types and while it does improve efficiency, the benefits are not nearly as high as the huge speed-up provided by perturbation theory.

Next: Public formulas

See Also
Fractal formulas