The Eigensolver (LOBPCG)

The ring

The board’s LOBPCG group draws one iteration as a ring, and the stage names there match this walkthrough exactly. The state is a block of BlockEntry { X, B·X, A·X }: three buffers per band, because caching the mass and operator images alongside the eigenvector block is what makes the rest cheap.

1. Residuals R = A·X − B·X·Λ come entirely from cached blocks: zero operator applies.
2. Lazy residual norms. Convergence is judged by eigenvalue movement, so the residual B-norms (which cost a block of mass applies) are computed only when a log line needs them. They are skipped on roughly 90% of iterations.
3. Deflation and soft locking project out converged bands and freeze them.
4. Preconditioning P = M⁻¹R: two or six FFTs per active band (operators).
5. Subspace assembly Z = [X, P, W]: current block, preconditioned residuals, and the previous search directions. Subspace dimension r ≈ 3 · bands.
6. SVQB B-orthonormalization: one Gram matrix and one dense eigendecomposition orthonormalize the whole subspace in a single shot. Near-dependent directions are rank-dropped, and the condition number is monitored (warnings at κ > 10¹⁰).
7. Batched A·Q: one operator apply per subspace vector. The FFT-heavy step.
8. Rayleigh-Ritz: form Aₛ = Qᴴ(AQ) by GEMM, solve the r×r Hermitian eigenproblem in f64. Both cross the precision boundary described in Precision & Memory.
9. Ritz update: X = Q·Y, B·X = BQ·Y, and crucially A·X = AQ·Y. Reusing the A·Q block from step 7 rebuilds the operator image by GEMM instead of a fresh block of operator applies. This one reuse pays for the entire caching scheme.

Warm starts collapse the iteration count

Within one band structure, k-points are solved in path order and each solve starts from the previous k-point’s eigenvectors, gauge-aligned by polar-decomposition parallel transport. The measured effect on this site’s own benchmark data: the first k-point needs around a dozen iterations, and warm-started k-points settle to a handful.

Press Run on the board to watch these numbers play out: the first k-point’s iterations are expanded stage by stage, and the remaining sweep runs at the real measured per-k wall times.