02/28/2009, 10:01 AM
(This post was last modified: 02/28/2009, 04:05 PM by sheldonison.)
sheldonison Wrote:I think there's a reasonable chance the higher derivatives will misbehave, and I think there's a reasonable chance the limit for the sexp limit may not even converge. The graph below is converting to base 1.6, from base 1.4453The results from base 1.45 to base 1.6 are very close to the results from base 1.44533, with the same 3rd derivative anomoly, and the same sexp function, consistent to about 3.0*10^-6. What this means is that even though the 3rd derivative misbehaves, there is still a chance that the sexp results has b approaches \( \eta \) may converge.
The wobble in the third derivative of the graph seems larger than the error term. I need to do some more error term estimates, and also see if converting from other bases (1.45 and 1.485, the other two spreadsheets I have generated), give similar results. ....
Even if the sexp limit definition converges, the higher derivatives for all bases will not have the property from Dimitrii's solution has, where the odd derivatives are positive for all values of x>-2. Also, an interesting set of graphs would be the wobble for the Dimitrii solution for base n, compared to the sexp generated for base n from a base approaching \( \eta \). Each base will have a wobble function. For base "e", the wobble has a peak to peak value of 0.0008, for smaller bases, the wobble gets smaller. For base=1.6, its about 0.00025, and for base 1.45, the relative wobble is about 0.000007. My theory is that the wobble for base conversions using Dimitrii's solutions becomes the anomoly in the third derivative for the sexp defined to have constant base conversions!
