Timeline for Derivative of adjoint action of exponential map
Current License: CC BY-SA 4.0
16 events
| when toggle format | what | by | license | comment | |
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| Nov 2, 2021 at 8:35 | vote | accept | Po C. | ||
| Nov 2, 2021 at 8:33 | answer | added | Po C. | timeline score: 0 | |
| Oct 11, 2021 at 1:51 | comment | added | Po C. | @LSpice I am actually doing numerical optimization that's why I need the derivative formula very badly. That simple formula has a great advantage that you can iteratively go higher order derivative. My codes is in finite dimensional of course, but indeed I have sparse linear operator of arbitrary dimension (maximum frequency for my choice of precision) in the first place. | |
| Oct 7, 2021 at 2:12 | comment | added | Michael Engelhardt | @LSpice - yes, that would be nice in general; for present purposes, since the OP states that "actually I just need finite-dimensional matrices", something more pedestrian might suffice ... | |
| Oct 6, 2021 at 23:42 | comment | added | LSpice | @MichaelEngelhardt, one problem with that is that it only makes sense after embedding in a matrix algebra. If we know that $X(t)$ and $Y$ lie in a prescribed Lie algebra, then it would be nice to have an expression intrinsic to the Lie algebra. (At least, I would find it nice! I don't know about @PoC.) | |
| Oct 6, 2021 at 21:28 | history | edited | Po C. | CC BY-SA 4.0 |
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| Oct 6, 2021 at 21:20 | comment | added | Po C. | I see. Maybe I should specify we may want something like $\frac{d}{dt}e^{X(t)} = e^{X(t)}\frac{1 - e^{-\mathrm{ad}_{X}}}{\mathrm{ad}_{X}}\frac{dX(t)}{dt}$ which has an explicit expansion. | |
| Oct 6, 2021 at 5:39 | history | edited | Po C. | CC BY-SA 4.0 |
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| Oct 6, 2021 at 5:33 | comment | added | Michael Engelhardt | Sure, that's what I mean when I say that "you know how to write down those two derivatives": Just insert the standard formula that you quote after your sentence "I start with the original formula for the derivative of the exponential map" | |
| Oct 6, 2021 at 5:18 | comment | added | Po C. | At least somethings based on $d/dt X(t)$ but not $d/dt \exp X(t)$. $d/dt \exp X(t)$ is not easy/obvious to compute/approximate numerically, for example. | |
| Oct 6, 2021 at 4:26 | comment | added | Michael Engelhardt | Well, there's the obvious answer $[\frac{d}{dt} e^X] Y e^{-X} + e^X Y [\frac{d}{dt} e^{-X}] $ (and you know how to write down those two derivatives) ... is that not an acceptable form? | |
| Oct 6, 2021 at 1:48 | history | edited | Po C. | CC BY-SA 4.0 |
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| Oct 6, 2021 at 0:00 | history | edited | Po C. | CC BY-SA 4.0 |
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| Oct 5, 2021 at 23:37 | history | edited | Po C. | CC BY-SA 4.0 |
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| Oct 5, 2021 at 23:13 | history | edited | LSpice | CC BY-SA 4.0 |
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| Oct 5, 2021 at 23:00 | history | asked | Po C. | CC BY-SA 4.0 |