Full Citation E. R. Davidson, "The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices", J. Comput. Phys. 17, 78-84 (1975). DOI: 10.1016/0021-9991(75)90065-0

Discussed on March 13, 2019.

Description

In this article, the author introduced a newly adapted method to accelerate large-scale configuration interaction calculations of electronic wavefunctions. It partially diminished some of the common drawbacks like the simultaneous change of all the cIs and the large consumptions of computational resource, while still maintain a reasonable accuracy as their result after testing the model.

Davidson diagonalization is a rapidly converging scheme which uses a subspace of eigenvectors of the given matrix to find its low-lying eigenvalues. Degeneracy of eigenvalues of the CI matrix can pose a problem for convergence for conventional methods, but not for the Davidson iterative diagonalization scheme. The matrix, if diagonally dominant, converges well. The iterative Davidson diagonalization procedure solves for a few lowest eigenvalues accurately, given a subset of eigenvectors of the corresponding higher-order matrix. Thus, the Davidson diagonalization scheme is a subspace expansion scheme (related: Krylov subspace expansion methods). Drawbacks: when initial guess vectors in the subspace are not close to actual eigenvectors, or are close to excited state eigenvectors (in that case, the lowest eigenvalue converged using this scheme might not be the actual ground state energy).

Supplementary reading resources:

• Joshua Goings: Davidson's Method

Keywords

CI; eigenvector; convergence; acceleration; iterative matrix diagonalization; Krylov subspace