Davidson Partial Eigensolver

A Rayleigh–Ritz iterative procedure for finding the k lowest eigenvalues and eigenvectors of a large matrix $\mathbf{A}$ without forming or storing it explicitly. Only matrix-vector products $\mathbf{Av}$ are required (supplied as a closure or via finite-difference gradient evaluations).

Algorithm

1. Initialise subspace $\mathbf{V}$ with $k$ unit vectors (or a user-supplied guess)
2. Build projected matrix $\tilde{\mathbf{A}}={\mathbf{V}}^{†}\mathbf{AV}$ (size $m\times m$, grows each step)
3. Diagonalise $\tilde{\mathbf{A}}\to$ Ritz values ${\theta }_i$ and Ritz vectors ${\mathbf{v}}_i=\mathbf{V}{\mathbf{e}}_i$
4. Check residuals ${\mathbf{r}}_i=\mathbf{A}{\mathbf{v}}_i-{\theta }_i{\mathbf{v}}_i$ for all $i<k$:
   • Converged when $\Vert {\mathbf{r}}_i\Vert \le \text{tol}\cdot \max (|{\theta }_i|,1)$ for all $i$
5. Expand: Orthogonalise the worst residual against $\mathbf{V}$ via Modified Gram-Schmidt (MGS); append the new column to $\mathbf{V}$ and $\mathbf{AV}$
6. Restart when $m\ge \text{max\_subspace}$: retain the $k$ best Ritz vectors

Modified Gram-Schmidt Orthogonalisation

Uses a sequential projection:

$$\mathbf{v}\leftarrow \mathbf{v}-(\mathbf{v}\cdot {\mathbf{w}}_j){\mathbf{w}}_j\text{for each basis column }{\mathbf{w}}_j$$ $$\mathbf{v}\leftarrow \mathbf{v}/\Vert \mathbf{v}\Vert$$

Returns None when the projected norm falls below tol (linear dependence).

Usage in OpenQuantum

The Davidson solver is used internally by refine_ts_hessian_via_davidson to determine an initial TS-mode direction from finite-difference Hessian-vector products, before the TS-BFGS update takes over.

It is also used by the Sella optimizer for adaptive eigenvalue correction.