Blocked-Davidson algorithm: Difference between revisions

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* Rayleigh-Ritz optimization ("subspace rotation") within the <math>3n_1</math> dimensional space spanned by <math>\{\psi^1/g^1/g^2\}</math>:
* Rayleigh-Ritz optimization ("subspace rotation") within the <math>3n_1</math> dimensional space spanned by <math>\{\psi^1/g^1/g^2\}</math>:
:<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, up to a depth <math>d</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:
:<math>\{\psi^1/g^1/g^2/../g^d\}</math
:<math>\{\psi^1/g^1/g^2/../g^{d-1}\}</math>
 
: Per default {{VASP}} will not iterate deeper than <math>d=4</math>.
* Continue iteration by adding a fourth set of preconditioned vectors if required, etc etc.
* Continue iteration by adding a fourth set of preconditioned vectors if required, etc etc.
* 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 18:03, 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 .
  • Continue iteration by adding a fourth set of preconditioned vectors if required, etc etc.
  • 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.