Eigenvalue decomposition derivatives needed

[riclarsson]

Hi,

This is somewhat of an open issue. An eigenvalue decomposition is necessary for line mixing ECS computations to be effective. This is done. We also need partial derivatives w.r.t. user input to fit with the rest of ARTS. I would like these derivatives to be analytical but I do not know how to take the partial derivatives of an eigenvalue decomposition. I need help here.

The specific function has been marked "FIXME: (Added 2021-01-19; Richard Larsson)" and is located in linemixing.cc at lines 13-58 at the merging of #281 . I can add computations of dW, dpop, and ddip if someone knows how to perform the required computations.

[nuitlejour]

will this help?
https://mathoverflow.net/questions/77057/partial-derivatives-of-eigen-value-decomposition-or-singular-value-decomposition

[riclarsson]

@nuitlejour Thank you for the suggestion! The answer pertains to real symmetrical matrices as far as I understand it. The matrix I need to compute is neither, so I do not believe it should apply. Sadly.

I am going to have a look through the suggested book, though this will take some time to dig through.

[nuitlejour]

use svd first, and the analytical derivatives on svd is quite simple, and does not require square or real matrix:
https://chat.openai.com/share/d76798ad-a00d-4e61-be19-c2905d20df90
and using the relationship between svd and eigenvalue decomposition:
https://en.wikipedia.org/wiki/Singular_value_decomposition
eigenvalue decomposition can be calculated.

[nuitlejour]

and also, I think in many cases, eigenvalue decomposition is not really needed, svd is more general.
maybe formulate 'line mixing ECS computations' with svd will be a better idea.

[dhjx1996]

Hi @riclarsson, Dion here, I stumbled upon this and I wanted to add that matrix calculus is well-defined (https://en.wikipedia.org/wiki/Matrix_calculus). Moreover, I have experience with it, in particular I learnt it from my Numerical Analysis class from forever ago, and can help with this.