qrisp.lanczos.regularize_S_H#

regularize_S_H(S: ArrayLike, H_mat: ArrayLike, cutoff: float = 0.01) → Tuple['ArrayLike', 'ArrayLike', 'ArrayLike'][source]#

Regularize the overlap matrix \(\mathbf{S}\) by retaining only eigenvectors with sufficiently large eigenvalues and project the Hamiltonian matrix \(\mathbf{H}\) accordingly.

This function applies a spectral cutoff: only directions in the Krylov subspace with eigenvalues above cutoff * max_eigenvalue are kept. Both the overlap matrix (\(\mathbf{S}\)) and the Hamiltonian matrix (\(\mathbf{H}\)) are projected onto this reduced subspace, ensuring numerical stability for subsequent generalized eigenvalue calculations. The regularized matrices are caculated as \(\tilde{S} = V^TSV\) and \(\tilde{H}=V^THV\) for a projection matrix \(V\).

Parameters:

SArrayLike, shape (D, D): The overlap matrix.
H_matArrayLike, shape (D, D): The Hamiltonian matrix.
cutofffloat: Eigenvalue threshold for regularizing \(\mathbf{S}\).

Returns:

S_regArrayLike, shape (D, D): The regularized overlap matrix.
H_regArrayLike, shape (D, D): The regularized Hamiltonian matrix in Krylov subspace.
VArrayLike, shape (D, D): The projection matrix.