Blocked-Davidson algorithm: Difference between revisions
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:<math>{\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k| k=1,..,n_1\}</math> | :<math>{\rm diag}\{\psi^1/g^1/g^2\} \Rightarrow \{ \psi^3_k| k=1,..,n_1\}</math> | ||
* If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace: | * If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace: | ||
:< | :<math>{\rm diag}\{\psi^1/g^1/g^2/../g^{d-1}\}\Rightarrow \{ \psi^d_k| k=1,..,n_1\}</math> | ||
: Per default {{VASP}} will not iterate deeper than <math>d=4</math>, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met. | : Per default {{VASP}} will not iterate deeper than <math>d=4</math>, though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met. | ||
* When the iteration is finished, store the optimized block of orbitals back in the set: | * When the iteration is finished, store the optimized block of orbitals back in the set: |
Revision as of 18:31, 19 October 2023
The workflow of the blocked-Davidson iterative matrix diagonalization scheme implemented in VASP is as follows:
- Take a subset (block) of orbitals out of the total set of NBANDS orbitals:
- .
- Extend the subspace spanned by by adding the preconditioned residual vectors of :
- Rayleigh-Ritz optimization ("subspace rotation") within the dimensional space spanned by , to determine the lowest eigenvectors:
- Extend the subspace with residuals of :
- Rayleigh-Ritz optimization ("subspace rotation") within the dimensional space spanned by :
- If need be the subspace may be extended by repetition of this cycle of adding residual vectors and Rayleigh-Ritz optimization of the resulting subspace:
- Per default VASP will not iterate deeper than , though it may break off even sooner when certain criteria that measure the convergence of the orbitals have been met.
- When the iteration is finished, store the optimized block of orbitals back in the set:
- .
- Continue with the next block .
- After each band has been optimized a Rayleigh-Ritz optimization in the complete subspace is performed.
This method is approximately a factor of 1.5-2 slower than RMM-DIIS, but always stable. It is available in parallel for any data distribution.