Blocked-Davidson algorithm

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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, up to a depth :
.
  • 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.