{"id":493,"date":"2008-06-15T16:00:51","date_gmt":"2008-06-15T23:00:51","guid":{"rendered":"https:\/\/www.reenigne.org\/blog\/?p=493"},"modified":"2008-05-23T16:34:03","modified_gmt":"2008-05-23T23:34:03","slug":"use-derivatives-for-soi","status":"publish","type":"post","link":"https:\/\/www.reenigne.org\/blog\/use-derivatives-for-soi\/","title":{"rendered":"Use derivatives for SOI"},"content":{"rendered":"<p>This is an elaboration on a point from <a href=\"https:\/\/www.reenigne.org\/blog\/recursive-subdivision\">an earlier blog post<\/a>.<\/p>\n<p>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<br \/>\nwell.<\/p>\n<p>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.<\/p>\n<p>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).<\/p>\n<p>It's also even easier to subdivide - instead of interpolating to find the intermediate iteration points, just evaluate the Taylor series.<\/p>\n<p>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.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[18],"tags":[],"class_list":["post-493","post","type-post","status-publish","format-standard","hentry","category-fractals"],"_links":{"self":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/493","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/comments?post=493"}],"version-history":[{"count":0,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/493\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/media?parent=493"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/categories?post=493"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/tags?post=493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}