Dear QE community,
I am new to ab initio program implementation. Recently I am studying the cg
solver in KS_Solvers/CG/ccgdiagg.f90. And I found the algorithm in the paper
https://arxiv.org/pdf/0906.2569 (QUANTUM ESPRESSO: a modular and open-source
software project for quantum simulations of materials), which published in
2009. I carefully read the Appendix A.2. Iterative diagonalization and formula
(A.8) to (A.22) which corresponds to conjugate gradient method. However, I am
struggling in understanding some pieces of code in the ccgdiagg.f90.
For the code in the No.164 to No.188 as listed here:
‘’’
!
! ... start iteration for this band
!
iterate: DO iter = 1, maxter
!
! ... calculate P (PHP)|y>
! ... ( P = preconditioning matrix, assumed diagonal )
!
g(:) = hpsi(:) / precondition(:)
ppsi(:) = spsi(:) / precondition(:)
!
! ... ppsi is now S P(P^2)|y> = S P^2|psi>)
!
es(1) = ddot( kdim2, spsi(1), 1, g(1), 1 )
es(2) = ddot( kdim2, spsi(1), 1, ppsi(1), 1 )
!
CALL mp_sum( es , intra_bgrp_comm )
!
es(1) = es(1) / es(2)
!
g(:) = g(:) - es(1) * ppsi(:)
!
! ... e1 = <y| S P^2 PHP|y> / <y| S S P^2|y> ensures that
! ... <g| S P^2|y> = 0
! ... orthogonalize to lowest eigenfunctions (already calculated)
!
! ... scg is used as workspace
!
‘’’
As we know that hpsi = H |psi> = H P |y>, since |y> = P^(-1) |psi> and P is
precondition matrix. Then g(:) = hpsi(:) / precondition(:) , seems imply g(:)
==> P^(-1) H P |y>, and similarly that ppsi(:) ==> P^(-1) S P |y>. Now the
question arrives, why the comment inside the code says that “! ... ppsi is
now S P(P^2)|y> = S P^2|psi>)” ?
And as for es(1) and es(2), it seems es(1) ==> <y| P S P^(-1) H P |y>, es(2)
==> <y| P S P^(-1) S P |y >. That are quiet different from that in the
comment “ e1 = <y| S P^2 PHP|y> / <y| S S P^2|y>”.
Is there any clue or reference materials to help understand that?
Sincerely,
HF Wang
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