Timeline for How can I distinguish a genuine solution of polynomial equations from a numerical near miss?
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 4, 2020 at 4:50 | answer | added | alpx | timeline score: 0 | |
| Sep 9, 2018 at 12:08 | comment | added | Qfwfq | @Kirill: ok thank you. I asked (besides for the name coincidence) because logicians' $\alpha$-theories have to do with infinitesimals, and I thought there might have been a slight possibility that this was linked with convergence rates or so. | |
| Sep 9, 2018 at 11:34 | comment | added | Kirill | @Qfwfq Knowing little about logic, I don't think so: Smale's $\alpha$-theory just gives a sufficient condition for Newton's method to converge from a given point (for an arbitrary set of nonlinear equations), giving also a strict bound on the distance from a true solution. The conditions can be checked numerically, with some difficulty, as they did in that alphaCertified paper I linked to. Certifying means checking that the numerical solution is close to its corresponding true solution, which is I think what the question is asking about. | |
| Sep 9, 2018 at 11:30 | comment | added | Qfwfq | Does Smale's alpha-theory (see @Kirill's comment above) have anything to do with logicians' $\alpha$-theories, or is it just a coincidence? | |
| Sep 8, 2018 at 23:46 | comment | added | arsmath | There are also all-solution homotopy methods for polynomial equations. Given that you have a very overdetermined system of equations, it might be worth trying an exact method such as homotopy, or Grobner bases, and see if you get lucky. | |
| Sep 8, 2018 at 23:04 | comment | added | Will Sawin | You could possibly apply numerical methods that some well-chosen 100 of the 1000 equations have a unique solution in some box. Then you could reduce the problem to exactly verifying that 101 equations have a joint solution, and that it lies in the box, 900 times. If the double-exponential in the number of variables is the main problem then this won't help much. | |
| S Sep 8, 2018 at 21:42 | history | suggested | Rodrigo de Azevedo |
Added tag.
|
|
| Sep 8, 2018 at 17:53 | review | Suggested edits | |||
| S Sep 8, 2018 at 21:42 | |||||
| Sep 2, 2018 at 18:23 | comment | added | RBega2 | Is there an answer for linear systems (i.e., the $f_j$ are affine functions) that doesn't reduce to solving the system? | |
| Sep 2, 2018 at 10:53 | comment | added | Kirill | Do you know about Smale's alpha-theory? (Like this: arxiv.org/abs/1011.1091) | |
| Sep 2, 2018 at 7:28 | answer | added | Federico Poloni | timeline score: 10 | |
| Sep 2, 2018 at 7:11 | history | edited | David Zhang | CC BY-SA 4.0 |
added 1 character in body; edited title
|
| Sep 2, 2018 at 6:56 | history | asked | David Zhang | CC BY-SA 4.0 |