Timeline for Numerical analytic continuation/asymptotics
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S May 14, 2023 at 19:08 | history | bounty ended | CommunityBot | ||
| S May 14, 2023 at 19:08 | history | notice removed | CommunityBot | ||
| May 11, 2023 at 7:27 | comment | added | lcv | @StevenClark the functions are solutions of a certain non linear ODE. For the purpose of this question only their Taylor series is available. | |
| May 11, 2023 at 4:11 | comment | added | Steven Clark | Could you please define the reference functions which you illustrate in blue in your four plots? | |
| May 10, 2023 at 18:12 | history | edited | lcv | CC BY-SA 4.0 |
Added another plot
|
| May 10, 2023 at 18:00 | comment | added | lcv | @fedja I added a couple of examples. In the second example it seems that the Taylor series converges to something different from the function after a certain point. | |
| May 10, 2023 at 17:55 | history | edited | lcv | CC BY-SA 4.0 |
Added two concrete examples
|
| May 10, 2023 at 3:10 | comment | added | fedja | "I have tried Padè approximants but they don't perform so well." Can you post the graphs of your Pade approximants? They are not really good for determining the value at $\infty$ directly but they should be reasonably good for finite values unless I misunderstand something, so if you are not going too far in $t$, you should have decent precision unless I miss some subtlety. | |
| May 7, 2023 at 8:22 | comment | added | Igor Khavkine | I don't know, but the locations of the poles will be part of your undetermined parameters. | |
| May 7, 2023 at 6:50 | comment | added | lcv | I see, I thought so. But would that work even without knowing the location of the poles? With some efforts I could obtain the location of their abscissa, but even that not very precisely. | |
| May 6, 2023 at 22:16 | comment | added | Igor Khavkine | It is the same idea as for Padé approximants. You start with undetermined parameters in your preferred representation (like Weierstrass or Mittag-Leffler) and expand at $z=0$. Then you have to see how the relation between the Taylor coefficients and undetermined parameters can be inverted. The problem may be underdetermined given only finite order Taylor data, in which case you'd need to make a guess for the higher order coefficients and estimate the error for that guess. | |
| May 6, 2023 at 21:47 | comment | added | lcv | Thanks for the comment. All I have of the function is it's Maclaurin series up to arbitrary order. Is there a way to go from the (truncated) series to the Weierstrass products or Mittag-Leffler series, even approximately? | |
| May 6, 2023 at 19:20 | comment | added | Igor Khavkine | Have you considered representing your function as a Weierstrass product (wiki) or a Mittag-Leffler series? These representations converge on the entire complex plane (minus the poles). Then it becomes a matter of matching the pole data to the Taylor series by expanding around the origin. | |
| S May 6, 2023 at 17:28 | history | bounty started | lcv | ||
| S May 6, 2023 at 17:28 | history | notice added | lcv | Draw attention | |
| May 4, 2023 at 16:27 | history | asked | lcv | CC BY-SA 4.0 |