Eigensolvers

eigh(A, *, max_sweeps=50, tol=1e-10)

Symmetric eigendecomposition A = V Λ Vᵀ for real symmetric A. Returns (eigenvalues, eigenvectors) with eigenvalues in ascending order.

When the NKI backend is active, dispatches to the Jacobi rotation kernel (currently scaffolded — falls back to torch.linalg.eigh until on-hardware validation lands). Pure-PyTorch path uses torch.linalg.eigh.

eigenvalues, eigenvectors = trnsolver.eigh(A)

eigh_generalized(A, B)

Generalized symmetric eigenproblem A x = λ B x with B SPD — the form that arises in the SCF iteration as F C = S C ε.

Reduces to standard form via Cholesky of B:

1. B = L Lᵀ
2. A' = L⁻¹ A L⁻ᵀ
3. eigh(A') → (Λ, V')
4. Back-transform V = L⁻ᵀ V'

eigenvalues, eigenvectors = trnsolver.eigh_generalized(F, S)

Backend notes

The Jacobi method is preferred over Householder tridiagonalization on Trainium because each Givens rotation maps cleanly to a 128×128 Tensor Engine tile. For details see the architecture notes.