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ED_FIT_GENERAL.f90
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ED_FIT_GENERAL.f90
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MODULE ED_FIT_GENERAL
USE SF_CONSTANTS
USE SF_OPTIMIZE, only:fmin_cg,fmin_cgminimize
USE SF_LINALG, only:eye,zeye,inv,inv_her,trace,operator(.x.) !BLAS xgemm operator overloading
USE SF_IOTOOLS, only:reg,free_unit,str
USE SF_ARRAYS, only:arange
USE SF_MISC, only:assert_shape
USE ED_INPUT_VARS
USE ED_VARS_GLOBAL
USE ED_AUX_FUNX
USE ED_BATH
USE ED_BATH_FUNCTIONS
implicit none
private
public :: chi2_fitgf_general
integer :: Ldelta
complex(8),dimension(:,:,:,:,:,:,:),allocatable :: FGmatrix
logical(8),dimension(:,:,:,:,:,:),allocatable :: Hmask
complex(8),dimension(:,:),allocatable :: Fdelta
real(8),dimension(:),allocatable :: Xdelta,Wdelta
integer :: totNorb,totNspin
integer,dimension(:),allocatable :: getIorb,getJorb,getIspin,getJspin,getIlat,getJlat
integer :: Orb_indx,Spin_indx,Spin_mask
integer,dimension(:),allocatable :: Nlambdas
!location of the maximum of the chisquare over Nlso.
integer :: maxchi_loc
!
type nsymm_vector
real(8),dimension(:),allocatable :: element
end type nsymm_vector
!
contains
!##################################################################
! THE CALCULATION OF THE \chi^2 FUNCTIONS USE PROCEDURES FURTHER
! BELOW TO EVALUATE INDEPENDENTLY THE ANDERSON MODEL:
! - DELTA,
! -\GRAD DELTA
! - G0
! THE LATTER ARE ADAPTED FROM THE PROCEDURES:
! DELTA_BATH_MATS
! GRAD_DELTA_BATH_MATS
! G0 BATH_MATS
! FOR, YOU NEED TO DECOMPOSE THE a INPUT ARRAY INTO ELEMENTS.
!##################################################################
!+-------------------------------------------------------------+
!PURPOSE : Chi^2 interface for GENERAL BATH
!+-------------------------------------------------------------+
subroutine chi2_fitgf_general(fg,bath_)
complex(8),dimension(:,:,:,:,:,:,:) :: fg ![Nlat][Nlat][Nspin][Nspin][Norb][Norb][Lmats]
real(8),dimension(:),intent(inout) :: bath_
real(8),dimension(:),allocatable :: array_bath
integer :: i,j,ilat,jlat,iorb,jorb,ispin,jspin,ibath,io,jo
integer :: iter,stride,counter,Asize
real(8) :: chi
logical :: check
character(len=256) :: suffix
integer :: unit
!
if(size(fg,1)/=Nlat) stop "chi2_fitgf_general error: size[fg,1]!=Nlat"
if(size(fg,2)/=Nlat) stop "chi2_fitgf_general error: size[fg,2]!=Nlat"
if(size(fg,3)/=Nspin)stop "chi2_fitgf_general error: size[fg,3]!=Nspin"
if(size(fg,4)/=Nspin)stop "chi2_fitgf_general error: size[fg,4]!=Nspin"
if(size(fg,5)/=Norb) stop "chi2_fitgf_general error: size[fg,5]!=Norb"
if(size(fg,6)/=Norb) stop "chi2_fitgf_general error: size[fg,6]!=Norb"
!
allocate(Nlambdas(Nbath))
!
check= check_bath_dimension(bath_)
if(.not.check)stop "chi2_fitgf_general error: wrong bath dimensions"
!
if(cg_pow/=2.AND.cg_norm=="frobenius")then
print *, "WARNING: CG_POW must be 2 for a meaningful definition of the Frobenius norm."
print *, " we'll still let you go ahead with the desired input, but please be "
print *, " be aware that CG_POW is not doing what you would expect for a chi^q"
endif
!
call allocate_dmft_bath()
call set_dmft_bath(bath_)
allocate(array_bath(size(bath_)-Nbath))
counter=0
do ibath=1,Nbath
counter=counter+1
Nlambdas(ibath)=NINT(bath_(counter))
enddo
array_bath=bath_(Nbath+1:size(bath_))
!
Ldelta = Lfit ; if(Ldelta>size(fg,7))Ldelta=size(fg,7)
!
!
allocate(FGmatrix(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta))
allocate(Hmask(Nlat,Nlat,Nspin,Nspin,Norb,Norb))
allocate(Xdelta(Ldelta))
allocate(Wdelta(Ldelta))
!
Xdelta = pi/beta*(2*arange(1,Ldelta)-1)
!
select case(cg_weight)
case default
Wdelta=1d0
case(2)
Wdelta=1d0*arange(1,Ldelta)
case(3)
Wdelta=Xdelta
end select
!
!
!----------------------------------------------------------------------------------------
!POSSIBLY TO BE USED AT SOME POINT (but beware that is meaningless for `FROBENIUS` norm)
Hmask=.true.
! Aren't we sure about FG_ij = FG_ji, coming from Hbath being hermitian?
!
! -> Hmask=Hbath_mask(wdiag=.false.,uplo=.true.) NO, this does more than
! that, putting .FALSE. where Hbath is zero, which may lead to tricky
! wrong fits: I have data showing this for a trimer, where at ~ 6.30d-10
! dmft error, hence a very well converged point, we have a nonnegligible
! Im(Weiss) component for ilat=1, jlat=3, despite Hmask(1,3) = Hmask(3,1)
! would result .FALSE. if computed with the Hbath_mask call above.
!
! For now I'd say that we'd better dump everything inside the \chi^2, hence Hmask=.true.,
! but we might want to consider exploiting hermiticity at least (probably not looking at
! Hbath though: who guarantees zeros therein imply zeros here?). Discussion needed...
!----------------------------------------------------------------------------------------
!
!
FGmatrix = fg
!
!
select case(cg_method) !0=NR-CG[default]; 1=CG-MINIMIZE
case default
if(cg_grad==0)then
#if __GNUC__ >= 8 || __INTEL_COMPILER >= 1500
write(LOGfile,*)" Using analytic gradient"
select case (cg_scheme)
case ("weiss")
call fmin_cg(array_bath,&
chi2_weiss_general,&
grad_chi2_weiss_general,&
iter,&
chi, &
itmax=cg_niter,&
ftol=cg_Ftol, &
istop=cg_stop, &
iverbose=(ed_verbose>3))
case ("delta")
call fmin_cg(array_bath,&
chi2_delta_general,&
grad_chi2_delta_general,&
iter,&
chi, &
itmax=cg_niter,&
ftol=cg_Ftol, &
istop=cg_stop, &
iverbose=(ed_verbose>3))
case default
stop "chi2_fitgf_general error: cg_scheme != [weiss,delta]"
end select
#else
STOP "analytic gradient not supported for gfortran < 8"
#endif
else
write(LOGfile,*)" Using numerical gradient"
select case (cg_scheme)
case ("weiss")
call fmin_cg(array_bath,chi2_weiss_general,&
iter,&
chi, &
itmax=cg_niter,&
ftol=cg_Ftol, &
istop=cg_stop, &
iverbose=(ed_verbose>3))
case ("delta")
call fmin_cg(array_bath,chi2_delta_general,&
iter,&
chi, &
itmax=cg_niter,&
ftol=cg_Ftol, &
istop=cg_stop, &
iverbose=(ed_verbose>3))
case default
stop "chi2_fitgf_general error: cg_scheme != [weiss,delta]"
end select
endif
!
case (1)
if(cg_grad==0)then
write(*,*) " "
write(*,*) "WARNING: analytic gradient not available with cg-method=1 (minimize f77 routine)"
write(*,*) " > we will force cg_grad=1 (so let the routine estimate the gradient) "
write(*,*) " "
endif
select case (cg_scheme)
case ("weiss")
call fmin_cgminimize(array_bath,chi2_weiss_general,&
iter,&
chi, &
itmax=cg_niter,&
ftol=cg_Ftol, &
new_version=cg_minimize_ver,&
hh_par=cg_minimize_hh, &
iverbose=(ed_verbose>3))
case ("delta")
call fmin_cgminimize(array_bath,chi2_delta_general,&
iter,&
chi, &
itmax=cg_niter,&
ftol=cg_Ftol, &
new_version=cg_minimize_ver,&
hh_par=cg_minimize_hh, &
iverbose=(ed_verbose>3))
case default
stop "chi2_fitgf_general error: cg_scheme != [weiss,delta]"
end select
!
end select
!
write(LOGfile,"(A,ES18.9,A,I5,A)")"chi^2|iter"//reg(ed_file_suffix)//'= ',chi," | ",iter," <-- All Orbs, All Spins"
!
suffix="_ALLorb_ALLspins"//reg(ed_file_suffix)
unit=free_unit()
open(unit,file="chi2fit_results"//reg(suffix)//".ed",position="append")
write(unit,"(ES18.9,1x,I5)") chi,iter
close(unit)
!
bath_(Nbath+1:size(bath_))=array_bath
call set_dmft_bath(bath_) ! *** bath_ --> dmft_bath *** (per write fit result)
call write_dmft_bath(LOGfile)
call save_dmft_bath()
!
call write_fit_result()
!
call get_dmft_bath(bath_) ! *** dmft_bath --> bath_ *** (bath in output)
call deallocate_dmft_bath()
deallocate(FGmatrix,Hmask,Xdelta,Wdelta)
deallocate(array_bath)
deallocate(Nlambdas)
!
contains
!
subroutine write_fit_result()
complex(8) :: fgand(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta)
integer :: i,j,s,l,ilat,jlat,iorb,jorb,ispin,jspin
real(8) :: w
if(cg_scheme=='weiss')then
fgand(:,:,:,:,:,:,:) = g0and_bath(xi*Xdelta(:))
else
fgand(:,:,:,:,:,:,:) = delta_bath(xi*Xdelta(:))
endif
!
do ilat=1,Nlat
do jlat=1,Nlat
do ispin=1,Nspin
do jspin=1,Nspin
do iorb=1,Norb
do jorb=1,Norb
suffix="_i"//reg(str(ilat))//&
"_j"//reg(str(jlat))//&
"_l"//reg(str(iorb))//&
"_m"//reg(str(jorb))//&
"_s"//reg(str(ispin))//&
"_r"//reg(str(jspin))//reg(ed_file_suffix)
unit=free_unit()
if(cg_scheme=='weiss')then
open(unit,file="fit_weiss"//reg(suffix)//".ed")
else
open(unit,file="fit_delta"//reg(suffix)//".ed")
endif
do i=1,Ldelta
w = Xdelta(i)
write(unit,"(5F24.15)")Xdelta(i),&
dimag(fg(ilat,jlat,ispin,jspin,iorb,jorb,i)),dimag(fgand(ilat,jlat,ispin,jspin,iorb,jorb,i)),&
dreal(fg(ilat,jlat,ispin,jspin,iorb,jorb,i)),dreal(fgand(ilat,jlat,ispin,jspin,iorb,jorb,i))
enddo
close(unit)
enddo
enddo
enddo
enddo
enddo
enddo
end subroutine write_fit_result
!
end subroutine chi2_fitgf_general
!##################################################################
! THESE PROCEDURES EVALUATE THE \chi^2 FUNCTIONS TO MINIMIZE.
!##################################################################
!
!
!+-----------------------------------------------------------------+
!PURPOSE: Evaluate the \chi^2 distance of \Delta_Anderson function.
!+-----------------------------------------------------------------+
function chi2_delta_general(a) result(chi2)
real(8),dimension(:) :: a
real(8) :: chi2
!
select case(cg_norm)
case ("elemental")
chi2 = chi2_delta_general_elemental(a)
case ("frobenius")
chi2 = chi2_delta_general_frobenius(a)
case default
stop "chi2_fitgf_general error: cg_norm != [elemental,frobenius]"
end select
!
end function chi2_delta_general
!
!
!> ELEMENTAL NORM: weighted sum over i\omega for each matrix element, then weighted sum over elements
function chi2_delta_general_elemental(a) result(chi2)
real(8),dimension(:) :: a
real(8) :: chi2
real(8),dimension(Ldelta) :: chi2_freq
real(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb) :: chi2_mtrx,Wmat
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: Delta
integer :: ilat,jlat,iorb,jorb,ispin,jspin
!
#ifdef _DEBUG
if(ed_verbose>5)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG chi2_delta_general_elemental. a:",a
#endif
!
Delta = delta_general(a)
!
do ilat=1,Nlat
do jlat=1,Nlat
do ispin=1,Nspin
do jspin=1,Nspin
do iorb=1,Norb
do jorb=1,Norb
chi2_freq = abs(Delta(ilat,jlat,ispin,jspin,iorb,jorb,1:Ldelta) - FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,1:Ldelta))
chi2_mtrx(ilat,jlat,ispin,jspin,iorb,jorb) = sum(chi2_freq**cg_pow/Wdelta) !Weighted sum over matsubara frqs
select case(cg_matrix)
case(0) !FLAT (all matrix elements weighted equal)
Wmat(ilat,jlat,ispin,jspin,iorb,jorb) = 1d0!0.25d0 !needs to depend on the hopping I think (\Delta=(D/2)^2*Gloc…)
case(1) !SPECTRAL (normalization through A(iw), element by element)
Wmat(ilat,jlat,ispin,jspin,iorb,jorb) = abs(sum(FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,1:Lmats)))/beta
#ifdef _DEBUG
if(ed_verbose>4)then
print*, "ilat: "//str(ilat)//" ispin: "//str(ispin)//" iorb: "//str(iorb)
print*, "jlat: "//str(jlat)//" jspin: "//str(jspin)//" jorb: "//str(jorb)
print*, "> Wmat: ", Wmat(ilat,jlat,ispin,jspin,iorb,jorb)/0.25d0
endif
#endif
end select
enddo
enddo
enddo
enddo
enddo
enddo
!
chi2 = sum(chi2_mtrx / Wmat, Hmask) !Weighted sum over matrix elements
chi2 = chi2 / Ldelta / count(Hmask) !Normalization over {iw} and Hmask
!
#ifdef _DEBUG
if(ed_verbose>3)write(Logfile,"(A,ES10.2)")"DEBUG chi2_delta_general_elemental. chi2:",chi2
#endif
!
end function chi2_delta_general_elemental
!
!
!> FROBENIUS NORM: global \chi^2 for all components, only i\omega are weighted
function chi2_delta_general_frobenius(a) result(chi2)
real(8),dimension(:) :: a
real(8) :: chi2
real(8),dimension(Ldelta) :: chi2_freq
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: Delta
complex(8),dimension(Nlat*Nspin*Norb,Nlat*Nspin*Norb) :: Delta_lso
integer :: l
!
#ifdef _DEBUG
if(ed_verbose>5)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG chi2_delta_general_frobenius. a:",a
#endif
!
Delta = delta_general(a)
!
do l=1,Ldelta
Delta_lso = nnn2lso_reshape(delta(:,:,:,:,:,:,l) - FGmatrix(:,:,:,:,:,:,l),Nlat,Nspin,Norb)
chi2_freq(l) = sqrt(trace(matmul(Delta_lso,conjg(transpose(Delta_lso)))))
enddo
!
chi2 = sum(chi2_freq**cg_pow/Wdelta) !Weighted sum over matsubara frqs
chi2 = chi2/Ldelta/(Nlat*Nspin*Norb) !Normalization over {iw} and Nlso
!
#ifdef _DEBUG
if(ed_verbose>3)write(Logfile,"(A,ES10.2)")"DEBUG chi2_delta_general_frobenius. chi2:",chi2
#endif
!
end function chi2_delta_general_frobenius
!
!
!+-------------------------------------------------------------+
!PURPOSE: Evaluate the \chi^2 distance of G0_Anderson function.
!+-------------------------------------------------------------+
function chi2_weiss_general(a) result(chi2)
real(8),dimension(:) :: a
real(8) :: chi2
!
select case(cg_norm)
case ("elemental")
chi2 = chi2_weiss_general_elemental(a)
case ("frobenius")
chi2 = chi2_weiss_general_frobenius(a)
case default
stop "chi2_fitgf_general error: cg_norm != [elemental,frobenius]"
end select
!
end function chi2_weiss_general
!
!
!> ELEMENTAL NORM: weighted sum over i\omega for each matrix element, then weighted sum over elements
function chi2_weiss_general_elemental(a) result(chi2)
real(8),dimension(:) :: a
real(8) :: chi2
real(8),dimension(Ldelta) :: chi2_freq
real(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb) :: chi2_mtrx,Wmat
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: g0and
integer :: ilat,jlat,iorb,jorb,ispin,jspin
!
#ifdef _DEBUG
if(ed_verbose>5)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG chi2_weiss_general_elemental. a:",a
#endif
!
g0and = g0and_general(a)
!
do ilat=1,Nlat
do jlat=1,Nlat
do ispin=1,Nspin
do jspin=1,Nspin
do iorb=1,Norb
do jorb=1,Norb
chi2_freq = abs(g0and(ilat,jlat,ispin,jspin,iorb,jorb,1:Ldelta) - FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,1:Ldelta))
chi2_mtrx(ilat,jlat,ispin,jspin,iorb,jorb) = sum(chi2_freq**cg_pow/Wdelta) !Weighted sum over matsubara frqs
select case(cg_matrix)
case(0) !FLAT (all matrix elements weighted equal)
Wmat(ilat,jlat,ispin,jspin,iorb,jorb) = 1d0
case(1) !SPECTRAL (normalization through A(iw), element by element)
Wmat(ilat,jlat,ispin,jspin,iorb,jorb) = abs(sum(FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,1:Lmats)))/beta
#ifdef _DEBUG
if(ed_verbose>4)then
print*, "ilat: "//str(ilat)//"ispin: "//str(ispin)//"iorb: "//str(iorb)
print*, "jlat: "//str(jlat)//"jspin: "//str(jspin)//"jorb: "//str(jorb)
print*, "> Wmat = ", Wmat(ilat,jlat,ispin,jspin,iorb,jorb)
endif
#endif
end select
enddo
enddo
enddo
enddo
enddo
enddo
!
chi2 = sum(chi2_mtrx / Wmat, Hmask) !Weighted sum over matrix elements
chi2 = chi2 / Ldelta / count(Hmask) !Normalization over {iw} and Hmask
!
#ifdef _DEBUG
if(ed_verbose>3)write(Logfile,"(A,ES10.2)")"DEBUG chi2_weiss_general_elemental. chi2:",chi2
#endif
!
end function chi2_weiss_general_elemental
!
!
!> FROBENIUS NORM: global \chi^2 for all components, only i\omega are weighted
function chi2_weiss_general_frobenius(a) result(chi2)
real(8),dimension(:) :: a
real(8) :: chi2
real(8),dimension(Ldelta) :: chi2_freq
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: g0and
complex(8),dimension(Nlat*Nspin*Norb,Nlat*Nspin*Norb) :: Delta_lso
integer :: l
!
#ifdef _DEBUG
if(ed_verbose>5)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG chi2_weiss_general_frobenius. a:",a
#endif
!
g0and = g0and_general(a)
!
do l=1,Ldelta
Delta_lso = nnn2lso_reshape(g0and(:,:,:,:,:,:,l) - FGmatrix(:,:,:,:,:,:,l),Nlat,Nspin,Norb)
chi2_freq(l) = sqrt(trace(matmul(Delta_lso,conjg(transpose(Delta_lso)))))
enddo
!
chi2 = sum(chi2_freq**cg_pow/Wdelta) !Weighted sum over matsubara frqs
chi2 = chi2/Ldelta/(Nlat*Nspin*Norb) !Normalization over {iw} and Nlso
!
#ifdef _DEBUG
if(ed_verbose>3)write(Logfile,"(A,ES10.2)")"DEBUG chi2_weiss_general_frobenius. chi2:",chi2
#endif
!
end function chi2_weiss_general_frobenius
!######################################################################
! THESE PROCEDURES EVALUATE THE >GRADIENTS< OF THE \chi^2 TO MINIMIZE.
!######################################################################
!
!
#if __GNUC__ >= 8 || __INTEL_COMPILER >= 1500
!+----------------------------------------------------------------------+
!PURPOSE: Evaluate the gradient \Grad\chi^2 of \Delta_Anderson function.
!+----------------------------------------------------------------------+
function grad_chi2_delta_general(a) result(dchi2)
real(8),dimension(:) :: a
real(8),dimension(size(a)) :: dchi2
!
select case(cg_norm)
case ("elemental")
dchi2 = grad_chi2_delta_general_elemental(a)
case ("frobenius")
dchi2 = grad_chi2_delta_general_frobenius(a)
case default
stop "chi2_fitgf_general error: cg_norm != [elemental,frobenius]"
end select
!
end function grad_chi2_delta_general
!
!
!> ELEMENTAL NORM: weighted sum over i\omega for each matrix element, then weighted sum over elements
function grad_chi2_delta_general_elemental(a) result(dchi2)
real(8),dimension(:) :: a
real(8),dimension(size(a)) :: dchi2
real(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,size(a)) :: df
real(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb) :: Wmat
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: Delta
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta,size(a)) :: dDelta
complex(8),dimension(Ldelta) :: Ftmp
real(8),dimension(Ldelta) :: Ctmp
integer :: ia,ilat,jlat,iorb,jorb,ispin,jspin
!
#ifdef _DEBUG
if(ed_verbose>5)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG grad_chi2_delta_general_elemental. a:",a
#endif
!
Delta = delta_general(a)
dDelta = grad_delta_general(a)
!
do ilat=1,Nlat
do jlat=1,Nlat
do ispin=1,Nspin
do jspin=1,Nspin
do iorb=1,Norb
do jorb=1,Norb
Ftmp = delta(ilat,jlat,ispin,jspin,iorb,jorb,1:Ldelta) - FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,1:Ldelta)
Ctmp = abs(Ftmp)**(cg_pow-2)
do ia = 1,size(a)
df(ilat,jlat,ispin,jspin,iorb,jorb,ia) = & !Weighted sum over matsubara frqs
sum( dreal(Ftmp) * dreal(dDelta(ilat,jlat,ispin,jspin,iorb,jorb,:,ia)) * Ctmp/Wdelta ) + &
sum( dimag(Ftmp) * dimag(dDelta(ilat,jlat,ispin,jspin,iorb,jorb,:,ia)) * Ctmp/Wdelta )
enddo
select case(cg_matrix)
case(0) !FLAT (all matrix elements weighted equal)
Wmat(ilat,jlat,ispin,jspin,iorb,jorb) = 1d0!0.25d0 !needs to depend on the hopping I think (\Delta=(D/2)^2*Gloc…)
case(1) !SPECTRAL (normalization through A(iw), element by element)
Wmat(ilat,jlat,ispin,jspin,iorb,jorb) = abs(sum(FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,1:Lmats)))/beta
end select
enddo
enddo
enddo
enddo
enddo
enddo
!
do ia=1,size(a)
dchi2(ia) = + cg_pow * sum( df(:,:,:,:,:,:,ia) / Wmat, Hmask) !Weighted sum over matrix elements
dchi2(ia) = dchi2(ia) / Ldelta / count(Hmask) !Normalization over {iw} and Hmask
enddo
!
#ifdef _DEBUG
if(ed_verbose>4)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG grad_chi2_delta_general_elemental. dChi2:",dchi2
#endif
!
end function grad_chi2_delta_general_elemental
!
!
!> FROBENIUS NORM: global \chi^2 for all components, only i\omega are weighted
function grad_chi2_delta_general_frobenius(a) result(dchi2)
real(8),dimension(:) :: a
real(8),dimension(size(a)) :: dchi2
real(8),dimension(Ldelta,size(a)) :: df
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: Delta
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta,size(a)) :: dDelta
complex(8),dimension(Ldelta) :: Ftmp
real(8),dimension(Ldelta,size(a)) :: dChi_freq
integer :: i,j,idelta,ilat,jlat,iorb,jorb,ispin,jspin
!
Delta = delta_general(a)
dDelta = grad_delta_general(a)
Ftmp=zero
df=zero
!
do idelta=1,Ldelta
do ilat=1,Nlat
do jlat=1,Nlat
do ispin=1,Nspin
do jspin=1,Nspin
do iorb=1,Norb
do jorb=1,Norb
!
Ftmp(idelta) = Ftmp(idelta) + abs(Delta(ilat,jlat,ispin,jspin,iorb,jorb,idelta)-FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,idelta))**2
do j=1,size(a)
df(idelta,j) = df(idelta,j) + &
real(Delta(ilat,jlat,ispin,jspin,iorb,jorb,idelta) - FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,idelta)) * &
real(dDelta(ilat,jlat,ispin,jspin,iorb,jorb,idelta,j)) + &
imag(Delta(ilat,jlat,ispin,jspin,iorb,jorb,idelta) - FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,idelta)) * &
imag(dDelta(ilat,jlat,ispin,jspin,iorb,jorb,idelta,j))
enddo
enddo
enddo
enddo
enddo
enddo
enddo
Ftmp(idelta) = cg_pow * (sqrt(Ftmp(idelta))**(cg_pow-2)) / Wdelta(idelta)
dchi_freq(idelta,:) = Ftmp(idelta) * df(idelta,:)
enddo
!
dchi2 = sum(dchi_freq,1)/Ldelta/(Nlat*Nspin*Norb)
!
#ifdef _DEBUG
if(ed_verbose>4)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG grad_chi2_delta_general_frobenius. dChi2:",dchi2
#endif
!
end function grad_chi2_delta_general_frobenius
!
!
!+------------------------------------------------------------------+
!PURPOSE: Evaluate the gradient \Grad\chi^2 of G0_Anderson function.
!+------------------------------------------------------------------+
function grad_chi2_weiss_general(a) result(dchi2)
real(8),dimension(:) :: a
real(8),dimension(size(a)) :: dchi2
!
select case(cg_norm)
case ("elemental")
dchi2 = grad_chi2_weiss_general_elemental(a)
case ("frobenius")
dchi2 = grad_chi2_weiss_general_frobenius(a)
case default
stop "chi2_fitgf_general error: cg_norm != [elemental,frobenius]"
end select
!
end function grad_chi2_weiss_general
!
!
!> ELEMENTAL NORM: weighted sum over i\omega for each matrix element, then weighted sum over elements
function grad_chi2_weiss_general_elemental(a) result(dchi2)
real(8),dimension(:) :: a
real(8),dimension(size(a)) :: dchi2
real(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,size(a)) :: df
real(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb) :: Wmat
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: g0and
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta,size(a)) :: dg0and
complex(8),dimension(Ldelta) :: Ftmp
real(8),dimension(Ldelta) :: Ctmp
integer :: ia,ilat,jlat,iorb,jorb,ispin,jspin
!
#ifdef _DEBUG
if(ed_verbose>5)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG grad_chi2_weiss_general_elemental. a:",a
#endif
!
g0and = g0and_general(a)
dg0and = grad_g0and_general(a)
!
do ilat=1,Nlat
do jlat=1,Nlat
do ispin=1,Nspin
do jspin=1,Nspin
do iorb=1,Norb
do jorb=1,Norb
Ftmp = g0and(ilat,jlat,ispin,jspin,iorb,jorb,1:Ldelta) - FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,1:Ldelta)
Ctmp = abs(Ftmp)**(cg_pow-2)
do ia = 1,size(a)
df(ilat,jlat,ispin,jspin,iorb,jorb,ia) = & !Weighted sum over matsubara frqs
sum( dreal(Ftmp) * dreal(dg0and(ilat,jlat,ispin,jspin,iorb,jorb,:,ia)) * Ctmp/Wdelta ) + &
sum( dimag(Ftmp) * dimag(dg0and(ilat,jlat,ispin,jspin,iorb,jorb,:,ia)) * Ctmp/Wdelta )
enddo
select case(cg_matrix)
case(0) !FLAT (all matrix elements weighted equal)
Wmat(ilat,jlat,ispin,jspin,iorb,jorb) = 1d0
case(1) !SPECTRAL (normalization through A(iw), element by element)
Wmat(ilat,jlat,ispin,jspin,iorb,jorb) = abs(sum(FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,1:Lmats)))/beta
end select
enddo
enddo
enddo
enddo
enddo
enddo
!
do ia=1,size(a)
dchi2(ia) = + cg_pow * sum( df(:,:,:,:,:,:,ia) / Wmat, Hmask) !Weighted sum over matrix elements
dchi2(ia) = dchi2(ia) / Ldelta / count(Hmask) !Normalization over {iw} and Hmask
enddo
!
#ifdef _DEBUG
if(ed_verbose>4)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG grad_chi2_weiss_general_elemental. dChi2:",dchi2
#endif
!
end function grad_chi2_weiss_general_elemental
!
!
!> FROBENIUS NORM: global \chi^2 for all components, only i\omega are weighted
function grad_chi2_weiss_general_frobenius(a) result(dchi2)
real(8),dimension(:) :: a
real(8),dimension(size(a)) :: dchi2
real(8),dimension(Ldelta,size(a)) :: df
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: g0and
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta,size(a)) :: dg0and
complex(8),dimension(Ldelta) :: Ftmp
real(8),dimension(Ldelta,size(a)) :: dChi_freq
integer :: i,j,idelta,ilat,jlat,iorb,jorb,ispin,jspin
!
!
print*, " "
print*, "WARNING: the analytic gradient of the Weiss field Frobenius distance "
print*, " has been found giving /QUALITATIVELY WRONG/ fitted spectra! "
print*, " > the issue has emerged in some Nlat=Nspin=Norb=1 test runs "
print*, " > while the bug is investigated please switch cg_scheme to "
print*, " 'delta' or cg_norm to 'elemental' (or give up analytic cg)"
print*, " "
!
!
g0and = g0and_general(a)
dg0and = grad_g0and_general(a)
Ftmp=zero
df=zero
!
do idelta=1,Ldelta
do ilat=1,Nlat
do jlat=1,Nlat
do ispin=1,Nspin
do jspin=1,Nspin
do iorb=1,Norb
do jorb=1,Norb
!
Ftmp(idelta) = Ftmp(idelta) + abs(g0and(ilat,jlat,ispin,jspin,iorb,jorb,idelta)-FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,idelta))**2
do j=1,size(a)
df(idelta,j) = df(idelta,j) + &
real(g0and(ilat,jlat,ispin,jspin,iorb,jorb,idelta) - FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,idelta)) * &
real(dg0and(ilat,jlat,ispin,jspin,iorb,jorb,idelta,j)) + &
imag(g0and(ilat,jlat,ispin,jspin,iorb,jorb,idelta) - FGmatrix(ilat,jlat,ispin,jspin,iorb,jorb,idelta)) * &
imag(dg0and(ilat,jlat,ispin,jspin,iorb,jorb,idelta,j))
enddo
enddo
enddo
enddo
enddo
enddo
enddo
Ftmp(idelta) = cg_pow * (sqrt(Ftmp(idelta))**(cg_pow-2)) / Wdelta(idelta)
dchi_freq(idelta,:) = Ftmp(idelta) * df(idelta,:)
enddo
!
dchi2 = sum(dchi_freq,1)/Ldelta/(Nlat*Nspin*Norb)
!
#ifdef _DEBUG
if(ed_verbose>4)write(Logfile,"(A,"//str(size(a))//"ES10.2)")"DEBUG grad_chi2_weiss_general_frobenius. dChi2:",dchi2
#endif
!
end function grad_chi2_weiss_general_frobenius
#endif
!##################################################################
! THESE PROCEDURES EVALUATE THE ANDERSON FUNCTIONS:
! - \Delta (hybridization)
! - g0 (weiss field)
!##################################################################
function delta_general(a) result(Delta)
real(8),dimension(:) :: a
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: Delta
integer :: ilat,jlat,ispin,jspin,iorb,jorb,ibath,isym
integer :: i,io,jo,ndx,stride
complex(8),dimension(Nlat*Nspin*Norb,Nlat*Nspin*Norb) :: V_k
complex(8),dimension(Nlat*Nspin*Norb,Nlat*Nspin*Norb) :: Haux
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb) :: invH_knnn
real(8),dimension(Nbath,Nlat*Nspin*Norb) :: dummy_Vbath
type(nsymm_vector),dimension(Nbath) :: dummy_lambda
complex(8) :: iw
!
!ACHTUNG! here the bath was a temporary one, since we removed the possibility to act on other baths we need to replicate the
!function behaviour. Rather ugly...
!Get Hs
stride = 0
do ibath=1,Nbath
allocate(dummy_lambda(ibath)%element(Nlambdas(ibath)))
!Get Vs
dummy_vbath(ibath,:) = a(stride+1:stride+Nlat*Nspin*Norb)
stride = stride + Nlat*Nspin*Norb
!get Lambdas
dummy_lambda(ibath)%element=a(stride+1:stride+Nlambdas(ibath))
stride=stride+Nlambdas(ibath)
enddo
!
!
Delta=zero
do i=1,Ldelta
iw = xi*Xdelta(i)
do ibath=1,Nbath
V_k = dzdiag(dummy_Vbath(ibath,:))
invH_knnn = Hbath_build(dummy_lambda(ibath)%element)
Haux = zeye(Nlat*Nspin*Norb)*iw - nnn2lso_reshape(invH_knnn,Nlat,Nspin,Norb)
call inv(Haux)
Haux = matmul(V_k,matmul(Haux,V_k))
invH_knnn = lso2nnn_reshape(Haux,Nlat,Nspin,Norb)
Delta(:,:,:,:,:,:,i) = Delta(:,:,:,:,:,:,i) + invH_knnn(:,:,:,:,:,:)
enddo
enddo
!
do ibath=1,Nbath
deallocate(dummy_lambda(ibath)%element)
enddo
!
end function delta_general
!
function g0and_general(a) result(G0and)
real(8),dimension(:) :: a
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: G0and,Delta
complex(8),dimension(Nlat*Norb*Nspin,Nlat*Norb*Nspin) :: zeta,fgorb
integer :: i,Nlso
integer :: ilat,jlat,iorb,jorb,ispin,jspin,io,jo
!
Nlso = Nlat*Norb*Nspin
!
Delta = delta_general(a)
!
do i=1,Ldelta
zeta = (xi*Xdelta(i)+xmu)*zeye(Nlso)
FGorb = zeta - nnn2lso_reshape(impHloc + Delta(:,:,:,:,:,:,i), Nlat,Nspin,Norb)
call inv(FGorb)
G0and(:,:,:,:,:,:,i) = lso2nnn_reshape(FGorb,Nlat,Nspin,Norb)
enddo
!
end function g0and_general
!
!
!
!
!
!
#if __GNUC__ >= 8 || __INTEL_COMPILER >= 1500
!##################################################################
! THESE PROCEDURES EVALUATE THE GRADIENT OF ANDERSON FUNCTIONS:
! - \Delta (hybridization)
! - g0 (weiss field)
!##################################################################
function grad_delta_general(a) result(dDelta)
real(8),dimension(:) :: a
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta,size(a)) :: dDelta
integer :: ilat,jlat,ispin,iorb,jorb,ibath
integer :: k,l,io,counter
complex(8),dimension(Nlat*Nspin*Norb,Nlat*Nspin*Norb) :: H_reconstructed,Htmp,Hbasis_lso,V_k
complex(8),dimension(Nlat*Nspin*Norb,Nlat*Nspin*Norb,Ldelta) :: Haux
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: invH_knn
real(8),dimension(Nbath,Nlat*Nspin*Norb) :: dummy_Vbath
type(nsymm_vector),dimension(Nbath) :: dummy_lambda
!
!
!Get Hs
counter = 0
do ibath=1,Nbath
if(allocated(dummy_lambda(ibath)%element))deallocate(dummy_lambda(ibath)%element)
allocate(dummy_lambda(ibath)%element(Nlambdas(ibath)))
!
dummy_vbath(ibath,:) = a(counter+1:counter+Nlat*Nspin*Norb)
counter = counter + Nlat*Nspin*Norb
!
dummy_lambda(ibath)%element=a(counter+1:counter+Nlambdas(ibath))
counter = counter + Nlambdas(ibath)
enddo
!
dDelta=zero
counter=0
!
do ibath=1,Nbath
V_k = dzdiag(dummy_Vbath(ibath,:))
H_reconstructed = nnn2lso_reshape(Hbath_build(dummy_lambda(ibath)%element),Nlat,Nspin,Norb)
do l=1,Ldelta
Haux(:,:,l) = zeye(Nlat*Nspin*Norb)*(xi*Xdelta(l)) - H_reconstructed
call inv(Haux(:,:,l))
!Derivate_Vp
do k = 1,Nlat*Nspin*Norb
Htmp(:,:) = zero
Htmp(:,k) = Htmp(:,k) + VH_prod(V_k,Haux(:,:,l),k)
Htmp(k,:) = Htmp(k,:) + HV_prod(V_k,Haux(:,:,l),k)
dDelta(:,:,:,:,:,:,l,counter+k) = lso2nnn_reshape(Htmp,Nlat,Nspin,Norb)
enddo
enddo
!
!counter = counter + Nlat*Nspin*Norb
!dDelta(:,:,:,:,:,:,:,counter) = invH_knn(:,:,:,:,:,:,:)
!
!Derivate_lambda_p
do k=1,Nlambdas(ibath)
counter = counter + 1
Hbasis_lso=nnn2lso_reshape(Hbath_basis(k)%O,Nlat,Nspin,Norb)
do l=1,Ldelta
Htmp = ((Haux(:,:,l) .x. Hbasis_lso)) .x. Haux(:,:,l)
Htmp = ((V_k .x. Htmp) .x. V_k)
dDelta(:,:,:,:,:,:,l,counter)=lso2nnn_reshape(Htmp,Nlat,Nspin,Norb)
enddo
enddo
enddo
!
do ibath=1,Nbath
if(allocated(dummy_lambda(ibath)%element))deallocate(dummy_lambda(ibath)%element)
enddo
!
end function grad_delta_general
!
!
function grad_g0and_general(a) result(dG0and)
real(8),dimension(:) :: a
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta,size(a)) :: dG0and,dDelta
complex(8),dimension(Nlat,Nlat,Nspin,Nspin,Norb,Norb,Ldelta) :: G0and
complex(8),dimension(Nlat*Nspin*Norb,Nlat*Nspin*Norb) :: dDelta_lso,dG0and_lso,G0and_lso
integer :: ilat,jlat,ispin,iorb,jorb
integer :: ik,l
!
G0and = g0and_general(a)
dDelta = grad_delta_general(a)
!
dG0and = zero
!
do l=1,Ldelta
G0and_lso=nnn2lso_reshape(g0and(:,:,:,:,:,:,l),Nlat,Nspin,Norb)
do ik=1,size(a)
dDelta_lso=nnn2lso_reshape(dDelta(:,:,:,:,:,:,l,ik),Nlat,Nspin,Norb)
dG0and_lso = (G0and_lso .x. dDelta_lso) .x. G0and_lso
dG0and(:,:,:,:,:,:,l,ik)=lso2nnn_reshape(dG0and_lso,Nlat,Nspin,Norb)
enddo
enddo
!
end function grad_g0and_general
#endif
! Auxiliary ℝ -> ℂ vector-to-diagonal-matrix constructor
function dzdiag(x) result(A)
real(8),dimension(:) :: x
complex(8),dimension(:,:),allocatable :: A
integer :: N,i
N=size(x,1)
allocate(A(N,N))
A=0.d0
do i=1,N
A(i,i)=x(i)
enddo
end function dzdiag
!Auxiliary functions for general derivative in V_k
function HV_prod(vv,HH,ig) result(AA)
complex(8),dimension(:,:) :: HH,vv
complex(8),dimension(:,:),allocatable :: Atmp
complex(8),dimension(:),allocatable :: AA
integer :: N,i,ig
N=size(vv,1)
allocate(Atmp(N,N),AA(N))
Atmp=matmul(HH,vv)
AA=Atmp(ig,:)
end function HV_prod
!
function VH_prod(vv,HH,ig) result(AA)