09/21/2015, 05:58 PM
(This post was last modified: 09/22/2015, 05:30 AM by sheldonison.)
(09/21/2015, 10:53 AM)tommy1729 Wrote: ....
In post 16 the fake derivatives depend on f '' alot.
But if f ' ' ' and f ' ' ' ' can be negative and a few other unfortunate properties then the second derivative might not always behave as you want.
... see if we can improve post 16 estimate with a f ' ' ' term !?
Although Sheldon argues that that term has little influence and for exp(x) it has NO influence.
Thanks for the comments Tommy. g''' is probably the error term; I don't know how the error term behaves, or how to approximate the error terms. For the exp "fake function", Stirling's approximation has a multiplicative error term of (1+1/12n), which is a fairly large slowly converging error term. I could try some numeric integrals to see how much the error comes down if the x^3 term is included.
\( \int_{-\infty i}^{\infty i} \exp(\frac{g''}{2}x^2)\;\; \) Gaussian approximation without the x^3 term
\( \int_{-\pi i}^{\pi i} \exp(\frac{g''}{2}x^2+\frac{g'''}{6}x^3)\;\; \) integral approximation with the x^3 term. For f(x)=exp(x), g'''=g''
There are other interesting cases. If f(x) is an even function with all odd derivatives=0, that is fine with me; even though this is different than Tommy's tpid#17. The interpolating fake(x) function will have all positive derivatives, filling in the "zero" derivatives to smooth out the function. Perhaps Tommy's suggestion is correct and fake functions work best when g'''(x)>=0? Also, what should be done for the Gaussian approximation when g''(x) is arbitrarily small??
\( a_n \approx \frac{\exp(g(h_n) - n h_n)}{\sqrt{2 \pi g''(h_n) }}\;\;\; \) if \( g''(h_n)<\frac{1}{2\pi} \), then the Gaussian approximation misbehaves; \( a_n \approx \exp(g(h_n) - n h_n) \)
- Sheldon
