08/22/2007, 10:41 AM
Wow, check this out. Using a 150-term truncation, I graphed the first 20 odd derivatives. I scaled each derivative so they would fit nicely on the same graph.
You can see the 20th odd derivative (i.e., the 39th derivative) is not quite symmetric, and it seems to backtrack a little. Up to that point, each odd derivative bottomed out a little to the right of the previous one, but the 20th odd derivative seems to bottom out a little to the left. Perhaps this is supposed to happen, but my hunch is that a few derivatives later we'll lose concavity, etc. I didn't save the data, so I can't get the next couple odd derivatives to check.
With only 50 terms, this breakdown happened several derivatives earlier (don't have the graph, I overwrote it accidently), followed by wild oscillations (similar to what my solution does at the 7th derivative). With only 30 terms, the breakdown occurs even earlier. Oddly enough, the 30-term truncation behaves better than my solution (if you don't count the fact that it has discontinuities in its higher derivatives at x=0, 1, e, etc.).
I'm currently calculating a 400-term trunctation to 2048 bits of precision, and I'm going to extract as many derivatives as the precision allows, unless and until the concavity of the odd derivatives breaks down. This will likely be an all-day calculation, but my curiosity must be satisfied...
You can see the 20th odd derivative (i.e., the 39th derivative) is not quite symmetric, and it seems to backtrack a little. Up to that point, each odd derivative bottomed out a little to the right of the previous one, but the 20th odd derivative seems to bottom out a little to the left. Perhaps this is supposed to happen, but my hunch is that a few derivatives later we'll lose concavity, etc. I didn't save the data, so I can't get the next couple odd derivatives to check.
With only 50 terms, this breakdown happened several derivatives earlier (don't have the graph, I overwrote it accidently), followed by wild oscillations (similar to what my solution does at the 7th derivative). With only 30 terms, the breakdown occurs even earlier. Oddly enough, the 30-term truncation behaves better than my solution (if you don't count the fact that it has discontinuities in its higher derivatives at x=0, 1, e, etc.).
I'm currently calculating a 400-term trunctation to 2048 bits of precision, and I'm going to extract as many derivatives as the precision allows, unless and until the concavity of the odd derivatives breaks down. This will likely be an all-day calculation, but my curiosity must be satisfied...
~ Jay Daniel Fox
