08/11/2009, 08:45 PM
(08/11/2009, 08:11 PM)bo198214 Wrote: If you have the interpolating polynomial, cant you compute the derivatives directly from it (instead of computing it numerically as it seems you do in the moment)?Sadly, no. If you will recall, terms calculated with the FMA will initially seem to converge, but then diverge to infinity. However, the terms can be calculated by iterative means, converging in a well-defined manner, so the sums exist. I suppose this might be a form of Ramanujan summation, but I'm not savvy enough to properly analyze this.
I suppose I could find the exact coefficients, but I must do so numerically, and this involves performing the same set of mathematical acrobatics: calculate a grid and numerically extract the derivatives. I can only get a few hundred, and I would need thousands to be practical, which unfortunately would require a super-computer. So I settle for interpolating when z is rather small (about 0.004 after 500 iterations), then iterate back up to where I need it to be. This is, of course, time consuming, but at least it's practical.
~ Jay Daniel Fox