Periodicity checking

Syntax:(periodicity bool)

Periodicity checking is one way to speed up the calculation. Areas inside the set always need ''maxiter'' iterations to determine that the point is probably inside the set (while it is rare for areas outside to need anywhere near that much). Often the orbital trajectory falls into a periodic, repeating cycle; if that can be detected, the calculation can be stopped early, as there’s no way that the orbit can ever leave the cycle again (hence it cannot diverge, hence the point must be inside the set).

Implementating this method efficiently is quite problematic. It slows down the cases where cycles are not found, because cycle-checking is quite hard work and has to take place for all points, even those that don’t become cyclic. Because of the inexactness of floating-point calculations, the cycles are never exact, so you need to use an error value. Higher error values mean that cycles will be detected sooner, while lower error values increase the exactness of the calculation. Higher values can introduce serious errors, especially at the front of the Mandelbrot set. XaoS detects this automatically and corrects for it in most cases, but sometimes it might be wrong. Also, other optimizations in XaoS (such as boundary tracing) don’t give this method much of a chance to run, since areas inside the set are usually not calculated at all.

That’s why the advantages of this optimization are questionable. You should probably experiment with enabling and disabling it. Sometimes XaoS is faster with this enabled, sometimes when disabled. Also, this method works only when incoloring methods are disabled, and only for some fractal types (some fractal types, e.g. newton, don’t have any concept of an area `inside the set’ at all.)

The tutorial chapter “Escape time fractals” has more information on fractal calculation in XaoS, and there is a lengthy section in the hacker’s manual (xaosdev.texinfo) devoted to the subject.

Available as: menu item, command line option, command