This is an elaboration on a point from an earlier blog post.
Synchronous orbit iteration is a way of speeding up the calculation of fractals. The idea comes from the observation that the orbits of nearby points follow similar trajectories for a while. So one can take a rectangular array of points and subdivide them once a rectangular array no longer approximates them
well.
It seems to me that a better way to do this might be to compute some the derivatives of the iteration function and iterate them instead of a grid of points, for the same reason that the accuracy of numerical integration is usually better improved by switching to a higher-order method than by decreasing the step size.
This method simplifies the algorithm which determines whether to subdivide or not (just see if the magnitude of the highest derivative exceeds some limit, rather than looking for rectangularity - which amounts to the same thing for the second derivative).
It's also even easier to subdivide - instead of interpolating to find the intermediate iteration points, just evaluate the Taylor series.
Of course, I'll need to do some experiments to determine if this is truly a better method (at the moment there are some more fundamental changes that my fractal plotter needs before I can play with this sort of thing). As far as I can tell nobody's ever tried it before, though. Classical SOI is difficult enough to get right.