Fix Hermitian Lanczos for complex matrices

[rahulgaur104]

Hi @pnkraemer,
Thank you for the nice package. I have been using matfree to solve a Hermitian eigenvalue stability problem. However, I realized that with a complex matrix, matfree requires some minor modifications for it to work correctly.

Summary

• use the conjugating inner product in Arnoldi/Hessenberg projections and norms
• symmetrize tridiagonal matrices with the Hermitian transpose
• update the direct Lanczos path and tests for complex Hermitian matrices

Tests

• python -m pytest (415 passed)

[rahulgaur104]

Closing this fork-based PR; I will try pushing a branch directly to the upstream repository instead.

[rahulgaur104]

Opening this fork-based PR, as I don't have permissions to create a separate branch.

[pnkraemer]

Thanks a lot for the PR! Really nice to make Lanczos and Arnoldi work for complex matrices.

I left some comments in the diff, would be nice to get them sorted before merging - but nothing major!

[rahulgaur104]

@pnkraemer
I think this is ready for review.

[rahulgaur104]

Unrelated but I used your package to write a dense and matrix-free shift invert solver here (page number 11).
If you want, we can add a shift invert solver to matfree.

[pnkraemer]

Unrelated but I used your package to write a dense and matrix-free shift invert solver here (page number 11). If you want, we can add a shift invert solver to matfree.

Super nice! Let's discuss this in the issue thread?

[pnkraemer]

Thanks a lot for that contribution! :)