08/11/2009, 08:03 PM

(08/11/2009, 05:16 PM)jaydfox Wrote: I then used the 129 points to calculate the first 128 derivatives. The last few derivatives have large errors (order of h^2, h^4, etc., where h is the grid spacing). I'm running a second calculation with a smaller grid spacing and more iterations, just to rule out numerical inaccuracies.I ran the second test, and it looks like the first run was pretty accurate. For the second test, I used 1000 iterations instead of 500, to minimize any imprecision in the interpolation polynomial (though I'm sure that even 200 iterations would suffice, but I'm being conservative here).

I used a grid spacing of 2^-128, or 20 bits smaller, so about 40 bits more accuracy for the 128th derivative, and considerably better accuracy for lower order derivatives.

When I compare the first results with the second, 128th terms of the Taylor series differ by about 10^-20, which is pretty incredible, considering that the 128th term is about 8.92033858123040e38. Thus, the accuracy was about one part in 10^59, for the -108 bit spacing, and thus considerably better for the -128 bit spacing.

Accuracy increases thereafter, so that most of the terms had about 300 decimal digits of accuracy when I was using the -108 bit spacing.

Unfortunately, this means my observation about the apparent radius of convergence was correct. It wasn't a fluke of the calculations. I'll try to identify a potential singularity, but assuming I can't (only 128 terms, after all), I'll try to get more terms.

I'll attach the coefficients shortly, if anyone is curious to see them.

~ Jay Daniel Fox