Timeline for How to solve this set of equations as efficiently as possible (with "efficiently" measured in FLOPS)?
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| Mar 26 at 15:21 | history | edited | gmvh | CC BY-SA 4.0 |
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| Mar 26 at 12:24 | history | edited | Simon | CC BY-SA 4.0 |
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| Jan 9, 2021 at 10:25 | history | edited | Simon | CC BY-SA 4.0 |
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| Jan 6, 2021 at 20:41 | comment | added | Simon | @FedericoPoloni Well, it converges in 50-100 iterations when the starting point is not that good. But I found a way to start with better initial guesses and that accelerated a bit further. I will have a look at the theoretical properties. | |
| Jan 6, 2021 at 18:38 | comment | added | Federico Poloni | @Simon Not that I know of. It might be easier to prove some theoretical properties, but I don't think it changes the picture significantly, if that fixed-point algorithm was already converging in a few iterations. | |
| Jan 6, 2021 at 15:15 | comment | added | Simon | @FedericoPoloni, is there somethin else I could look at now that I have corrected that A is positive only? | |
| Jan 5, 2021 at 11:05 | comment | added | David Roberts♦ | @Federico I'll take that on board. | |
| Jan 5, 2021 at 9:29 | history | edited | Simon | CC BY-SA 4.0 |
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| Jan 5, 2021 at 9:24 | history | edited | Simon | CC BY-SA 4.0 |
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| Jan 4, 2021 at 13:08 | comment | added | Federico Poloni | I'd suggest to write the same information as an answer instead, even without adding details. In this way it can be accepted, for instance, downvoted if it's wrong, searched using the MO search box, edited for corrections (for instance automated fixes to dead links), etc. There are many details that simply don't work with comments. | |
| Jan 4, 2021 at 10:11 | comment | added | David Roberts♦ | @Federico Mostly because I don't care about gaining rep points, and I would spend too long writing a detailed answer, more than I should. It's either give some information in a comment, or none at all. | |
| Jan 4, 2021 at 8:14 | comment | added | Federico Poloni | ...and why do you give these hints or outline answers as comments, exactly? I don't know where the practice started, but it seems a misuse of the system to me. These "hints" cannot be downvoted even when they are completely wrong. | |
| Jan 4, 2021 at 1:11 | comment | added | David Roberts♦ | @Federico it's the difference between a 16k+ rep user telling someone with <1k rep not to be afraid of downvotes, and one of the most respected algebraic geometers of our time deciding how he wishes to voluntarily impart his deep understanding. I know why he so chose but I don't know if it's public knowledge so I'm not going to share it here. I also choose to sometimes give hints or outline answers in comments, and not because I'm afraid of downvotes. | |
| Jan 3, 2021 at 21:03 | comment | added | Federico Poloni | @DavidRoberts Well, "someone else does that, too" does not sound like a convincing argument, bud I'd be interested to learn the reasons behind that choice. | |
| Jan 3, 2021 at 20:54 | comment | added | David Roberts♦ | @Federico tell that to Brian Conrad, who started only ever answering questions in comments, deliberately. | |
| Jan 3, 2021 at 11:28 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
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| Jan 3, 2021 at 10:46 | comment | added | Federico Poloni | @ChristopherA.Wong Comments are meant to suggest improvements to the question. Yours is a perfectly fine answer, in my view, not a comment. Posting this kind of content as comments defeats the purpose of the reputation and upvotes system, and I think it is bad practice on SE. | |
| Jan 3, 2021 at 10:35 | comment | added | Christopher A. Wong | @FedericoPoloni I don't think my comment is sufficiently detailed, nor backed by sufficient mathematical or computational evidence, to be an answer at this point. | |
| Jan 3, 2021 at 10:26 | comment | added | Federico Poloni | Using successive substitution with damping I get all of them solved uniquely. What do you mean? That method returns one solution; of course it "solves them uniquely". However, what happens if you run the methods many times with different starting points? Does it always converge to the same solution? Or do you get multiple ones? | |
| Jan 3, 2021 at 10:24 | comment | added | Federico Poloni | @ChristopherA.Wong Please don't post answers in comments. Show that you are not afraid of downvotes and use the answer box. :) | |
| Jan 3, 2021 at 10:23 | comment | added | Simon | @ChristopherA.Wong In some cases I am already using good initial guesses and it workd quite well. But for some cases I have no initial guess. | |
| Jan 3, 2021 at 10:23 | answer | added | Federico Poloni | timeline score: 2 | |
| Jan 3, 2021 at 10:20 | comment | added | Simon | @FedericoPoloni I am an engineer that needs to solve this set of equations for many A with n ranging from 100-1000. Using successive substitution with damping I get all of them solved uniquely. I am however interested if it is possible to solve them more efficiently as I need to solve millions of them. | |
| Jan 3, 2021 at 10:18 | comment | added | Christopher A. Wong | For large systems, Newton methods will be very poor until you are fairly close to the minimizing point. My suspicion is that the fastest method will look (1) good heuristic for initial guess, (2) fast first-order method for a few steps, and then (3) Newton iterations till convergence. Due to your additional structure it might be possible to do more rigorously than this, of course, but generally it's pretty difficult to rigorously analyze such nonlinear systems. | |
| Jan 3, 2021 at 10:09 | comment | added | Federico Poloni | Are you looking for a specific solution, or any one goes? Or have you proved that the solution is unique? (Because, in general, it won't.) | |
| Jan 3, 2021 at 9:50 | review | First posts | |||
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| Jan 3, 2021 at 9:43 | history | asked | Simon | CC BY-SA 4.0 |