How to obtain left eigenvectors in zndrv1 for non-Hermitian problems?

[Jiangsese]

Hi, I’m currently working on non-Hermitian quantum many-body systems using the ARPACK routine zndrv1, which is built upon znaupd and zneupd. I’ve encountered a conceptual issue and would like to ask for clarification:

When solving non-Hermitian eigenvalue problems, we need both the left and right eigenvectors to correctly evaluate observables such as $⟨\psi_L|O|\psi_R⟩$.

However, as far as I understand, zndrv1 (and more generally, zneupd) only returns the right eigenvectors (Ritz vectors). So I would like to ask:

1. Is there any way to configure zneupd (e.g., via the howmny, rvec, or other parameters) to directly return left eigenvectors?
2. Or is it necessary to run a second ARPACK calculation using the adjoint matrix ($ H^\dagger$ ) — that is, replacing the matrix-vector product with ( $H^\dagger$ v ) — in order to extract the left eigenvectors?
3. Is there any official documentation, manual, or example that explain explicitly for the process of how to calculate the left eigenvectors?

Thank you very much for your help!

[fghoussen]

Hi @Jiangsese, arpack is maintained-only: the doc/code dates from the 70's or so!... Not always clear how to find it way through it...

If you find your answer, do not hesitate to PR an update of the FAQ section from the README! :)

Is there any way to configure zneupd (e.g., via the howmny, rvec, or other parameters) to directly return left eigenvectors?

I guess no

Or is it necessary to run a second ARPACK calculation using the adjoint matrix ($ H^\dagger$ ) — that is, replacing the matrix-vector product with ( H † v ) — in order to extract the left eigenvectors?

I guess yes

Is there any official documentation, manual, or example that explain explicitly for the process of how to calculate the left eigenvectors?

I guess no: the best doc you'll find is referenced in the FAQ https://github.com/opencollab/arpack-ng/tree/master?tab=readme-ov-file#faq (ARPACK user's guide)