2019-01-25 11:39:31 +01:00
|
|
|
subroutine svd(A,LDA,U,LDU,D,Vt,LDVt,m,n)
|
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Compute A = U.D.Vt
|
|
|
|
!
|
|
|
|
! LDx : leftmost dimension of x
|
|
|
|
!
|
2020-11-11 15:51:19 +01:00
|
|
|
! Dimension of A is m x n
|
2019-01-25 11:39:31 +01:00
|
|
|
!
|
|
|
|
END_DOC
|
|
|
|
|
|
|
|
integer, intent(in) :: LDA, LDU, LDVt, m, n
|
|
|
|
double precision, intent(in) :: A(LDA,n)
|
2020-09-30 20:48:27 +02:00
|
|
|
double precision, intent(out) :: U(LDU,min(m,n))
|
2019-01-25 11:39:31 +01:00
|
|
|
double precision,intent(out) :: Vt(LDVt,n)
|
|
|
|
double precision,intent(out) :: D(min(m,n))
|
|
|
|
double precision,allocatable :: work(:)
|
|
|
|
integer :: info, lwork, i, j, k
|
|
|
|
|
|
|
|
double precision,allocatable :: A_tmp(:,:)
|
|
|
|
allocate (A_tmp(LDA,n))
|
2020-12-23 02:42:38 +01:00
|
|
|
do k=1,n
|
|
|
|
do i=1,m
|
|
|
|
A_tmp(i,k) = A(i,k)
|
|
|
|
enddo
|
|
|
|
enddo
|
2019-01-25 11:39:31 +01:00
|
|
|
|
|
|
|
! Find optimal size for temp arrays
|
|
|
|
allocate(work(1))
|
|
|
|
lwork = -1
|
2021-01-12 14:23:33 +01:00
|
|
|
call dgesvd('S','S', m, n, A_tmp, LDA, &
|
2019-01-25 11:39:31 +01:00
|
|
|
D, U, LDU, Vt, LDVt, work, lwork, info)
|
2019-11-21 09:56:30 +01:00
|
|
|
! /!\ int(WORK(1)) becomes negative when WORK(1) > 2147483648
|
2021-01-12 14:23:33 +01:00
|
|
|
lwork = max(int(work(1)), 10*MIN(M,N))
|
2019-01-25 11:39:31 +01:00
|
|
|
deallocate(work)
|
|
|
|
|
|
|
|
allocate(work(lwork))
|
2021-01-12 14:23:33 +01:00
|
|
|
call dgesvd('S','S', m, n, A_tmp, LDA, &
|
2019-01-25 11:39:31 +01:00
|
|
|
D, U, LDU, Vt, LDVt, work, lwork, info)
|
2020-11-11 15:51:19 +01:00
|
|
|
deallocate(A_tmp,work)
|
2019-01-25 11:39:31 +01:00
|
|
|
|
|
|
|
if (info /= 0) then
|
|
|
|
print *, info, ': SVD failed'
|
|
|
|
stop
|
|
|
|
endif
|
|
|
|
|
2020-12-23 02:45:20 +01:00
|
|
|
do j=1,min(m,n)
|
2020-11-11 15:51:19 +01:00
|
|
|
do i=1,m
|
|
|
|
if (dabs(U(i,j)) < 1.d-14) U(i,j) = 0.d0
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
do j=1,n
|
|
|
|
do i=1,n
|
|
|
|
if (dabs(Vt(i,j)) < 1.d-14) Vt(i,j) = 0.d0
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
end
|
|
|
|
|
|
|
|
subroutine svd_symm(A,LDA,U,LDU,D,Vt,LDVt,m,n)
|
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Compute A = U.D.Vt
|
|
|
|
!
|
|
|
|
! LDx : leftmost dimension of x
|
|
|
|
!
|
|
|
|
! Dimension of A is m x n
|
|
|
|
!
|
|
|
|
END_DOC
|
|
|
|
|
|
|
|
integer, intent(in) :: LDA, LDU, LDVt, m, n
|
|
|
|
double precision, intent(in) :: A(LDA,n)
|
|
|
|
double precision, intent(out) :: U(LDU,min(m,n))
|
|
|
|
double precision,intent(out) :: Vt(LDVt,n)
|
|
|
|
double precision,intent(out) :: D(min(m,n))
|
|
|
|
double precision,allocatable :: work(:)
|
|
|
|
integer :: info, lwork, i, j, k
|
|
|
|
|
|
|
|
double precision,allocatable :: A_tmp(:,:)
|
|
|
|
allocate (A_tmp(LDA,n))
|
|
|
|
A_tmp(:,:) = A(:,:)
|
|
|
|
|
|
|
|
! Find optimal size for temp arrays
|
|
|
|
allocate(work(1))
|
|
|
|
lwork = -1
|
|
|
|
call dgesvd('A','A', m, n, A_tmp, LDA, &
|
|
|
|
D, U, LDU, Vt, LDVt, work, lwork, info)
|
|
|
|
! /!\ int(WORK(1)) becomes negative when WORK(1) > 2147483648
|
|
|
|
lwork = max(int(work(1)), 5*MIN(M,N))
|
|
|
|
deallocate(work)
|
|
|
|
|
|
|
|
allocate(work(lwork))
|
|
|
|
call dgesvd('A','A', m, n, A_tmp, LDA, &
|
|
|
|
D, U, LDU, Vt, LDVt, work, lwork, info)
|
|
|
|
deallocate(A_tmp,work)
|
|
|
|
|
|
|
|
if (info /= 0) then
|
|
|
|
print *, info, ': SVD failed'
|
|
|
|
stop
|
|
|
|
endif
|
|
|
|
|
|
|
|
! Iterative refinement
|
|
|
|
! --------------------
|
|
|
|
! https://doi.org/10.1016/j.cam.2019.112512
|
|
|
|
|
|
|
|
integer :: iter
|
|
|
|
double precision,allocatable :: R(:,:), S(:,:), T(:,:)
|
|
|
|
double precision,allocatable :: sigma(:), F(:,:), G(:,:)
|
|
|
|
double precision :: alpha, beta, x, thresh
|
|
|
|
|
|
|
|
allocate (R(m,m), S(n,n), T(m,n), sigma(m), F(m,m), G(n,n), A_tmp(m,n))
|
|
|
|
sigma = 0.d0
|
|
|
|
R = 0.d0
|
|
|
|
S = 0.d0
|
|
|
|
T = 0.d0
|
|
|
|
F = 0.d0
|
|
|
|
G = 0.d0
|
|
|
|
|
|
|
|
thresh = 1.d-8
|
|
|
|
call restore_symmetry(m,m,U,size(U,1),thresh)
|
|
|
|
call restore_symmetry(n,n,Vt,size(Vt,1),thresh)
|
|
|
|
|
|
|
|
do iter=1,4
|
|
|
|
do k=1,n
|
|
|
|
A_tmp(1:m,k) = D(k) * U(1:m,k)
|
|
|
|
enddo
|
|
|
|
|
|
|
|
call dgemm('N','N',m,n,n,1.d0,A_tmp,size(A_tmp,1), &
|
|
|
|
Vt,size(Vt,1),0.d0,R,size(R,1))
|
|
|
|
|
|
|
|
print *, maxval(dabs(R(1:m,1:n) - A(1:m,1:n)))
|
|
|
|
|
|
|
|
|
|
|
|
call dgemm('T','N',m,m,m,-1.d0,U,size(U,1), &
|
|
|
|
U,size(U,1),0.d0,R,size(R,1))
|
|
|
|
do i=1,m
|
|
|
|
R(i,i) = R(i,i) + 1.d0
|
|
|
|
enddo
|
|
|
|
|
|
|
|
call dgemm('N','T',n,n,n,-1.d0,Vt,size(Vt,1), &
|
|
|
|
Vt,size(Vt,1),0.d0,S,size(S,1))
|
|
|
|
do i=1,n
|
|
|
|
S(i,i) = S(i,i) + 1.d0
|
|
|
|
enddo
|
|
|
|
|
|
|
|
|
|
|
|
call dgemm('T','N',m,n,m,1.d0,U,size(U,1), &
|
|
|
|
A,size(A,1),0.d0,A_tmp,size(A_tmp,1))
|
|
|
|
|
|
|
|
call dgemm('N','T',m,n,n,1.d0,A_tmp,size(A_tmp,1), &
|
|
|
|
Vt,size(Vt,1),0.d0,T,size(T,1))
|
|
|
|
|
|
|
|
|
|
|
|
do i=1,n
|
|
|
|
sigma(i) = T(i,i)/(1.d0 - (R(i,i)+S(i,i))*0.5d0)
|
|
|
|
F(i,i) = 0.5d0*R(i,i)
|
|
|
|
G(i,i) = 0.5d0*S(i,i)
|
|
|
|
enddo
|
|
|
|
|
|
|
|
do j=1,n
|
|
|
|
do i=1,n
|
|
|
|
if (i == j) cycle
|
|
|
|
alpha = T(i,j) + sigma(j) * R(i,j)
|
|
|
|
beta = T(i,j) + sigma(j) * S(i,j)
|
|
|
|
x = 1.d0 / (sigma(j)*sigma(j) - sigma(i)*sigma(i))
|
|
|
|
F(i,j) = (alpha * sigma(j) + beta * sigma(i)) * x
|
|
|
|
G(i,j) = (alpha * sigma(i) + beta * sigma(j)) * x
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
do i=1,n
|
|
|
|
x = 1.d0/sigma(i)
|
|
|
|
do j=n+1,m
|
|
|
|
F(i,j) = -T(j,i) * x
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
do i=n+1,m
|
|
|
|
do j=1,n
|
|
|
|
F(i,j) = R(i,j) - F(j,i)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
do i=n+1,m
|
|
|
|
do j=n+1,m
|
|
|
|
F(i,j) = R(i,j)*0.5d0
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
D(1:min(n,m)) = sigma(1:min(n,m))
|
|
|
|
call dgemm('N','N',m,m,m,1.d0,U,size(U,1),F,size(F,1), &
|
|
|
|
0.d0, A_tmp, size(A_tmp,1))
|
|
|
|
do j=1,m
|
|
|
|
do i=1,m
|
|
|
|
U(i,j) = U(i,j) + A_tmp(i,j)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
call dgemm('T','N',n,n,n,1.d0,G,size(G,1),Vt,size(Vt,1), &
|
|
|
|
0.d0, A_tmp, size(A_tmp,1))
|
|
|
|
do j=1,n
|
|
|
|
do i=1,n
|
|
|
|
Vt(i,j) = Vt(i,j) + A_tmp(i,j)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
thresh = 0.01d0 * thresh
|
|
|
|
call restore_symmetry(m,m,U,size(U,1),thresh)
|
|
|
|
call restore_symmetry(n,n,Vt,size(Vt,1),thresh)
|
|
|
|
|
|
|
|
enddo
|
|
|
|
|
|
|
|
deallocate(A_tmp,R,S,F,G,sigma)
|
2019-01-25 11:39:31 +01:00
|
|
|
end
|
|
|
|
|
2020-09-30 20:48:27 +02:00
|
|
|
subroutine eigSVD(A,LDA,U,LDU,D,Vt,LDVt,m,n)
|
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Algorithm 3 of https://arxiv.org/pdf/1810.06860.pdf
|
|
|
|
!
|
|
|
|
! A(m,n) = U(m,n) D(n) Vt(n,n) with m>n
|
|
|
|
END_DOC
|
|
|
|
integer, intent(in) :: LDA, LDU, LDVt, m, n
|
|
|
|
double precision, intent(in) :: A(LDA,n)
|
|
|
|
double precision, intent(out) :: U(LDU,n)
|
|
|
|
double precision,intent(out) :: Vt(LDVt,n)
|
|
|
|
double precision,intent(out) :: D(n)
|
|
|
|
|
|
|
|
integer :: i,j,k
|
|
|
|
|
|
|
|
if (m<n) then
|
|
|
|
stop -1
|
|
|
|
call svd(A,LDA,U,LDU,D,Vt,LDVt,m,n)
|
|
|
|
return
|
|
|
|
endif
|
|
|
|
|
|
|
|
double precision, allocatable :: B(:,:), V(:,:)
|
|
|
|
allocate(B(n,n))
|
|
|
|
! B = - At . A
|
|
|
|
call dgemm('T','N',n,n,m,-1.d0,A,size(A,1),A,size(A,1),0.d0,B,size(B,1))
|
|
|
|
|
|
|
|
! V, D = eig(B)
|
|
|
|
allocate(V(n,n))
|
|
|
|
call lapack_diagd(D,V,B,n,n)
|
|
|
|
deallocate(B)
|
|
|
|
do j=1,n
|
|
|
|
do i=1,n
|
|
|
|
Vt(i,j) = V(j,i)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
! S = sqrt(-D)
|
|
|
|
! U = A.V.S^-1
|
|
|
|
! U = A.(S^-1.vt)t
|
|
|
|
|
|
|
|
do k=1,n
|
|
|
|
if (D(k) >= 0.d0) then
|
|
|
|
exit
|
|
|
|
endif
|
|
|
|
D(k) = dsqrt(-D(k))
|
|
|
|
call dscal(n, 1.d0/D(k), V(1,k), 1)
|
|
|
|
enddo
|
|
|
|
D(k:n) = 0.d0
|
|
|
|
k=k-1
|
|
|
|
call dgemm('N','N',m,n,k,1.d0,A,size(A,1),V,size(V,1),0.d0,U,size(U,1))
|
|
|
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
|
|
|
subroutine randomized_svd(A,LDA,U,LDU,D,Vt,LDVt,m,n,q,r)
|
|
|
|
implicit none
|
|
|
|
include 'constants.include.F'
|
|
|
|
BEGIN_DOC
|
|
|
|
! Randomized SVD: rank r, q power iterations
|
|
|
|
!
|
|
|
|
! 1. Sample column space of A with P: Z = A.P where P is random with r+p columns.
|
|
|
|
!
|
|
|
|
! 2. Power iterations : Z <- X . (Xt.Z)
|
|
|
|
!
|
|
|
|
! 3. Z = Q.R
|
|
|
|
!
|
|
|
|
! 4. Compute SVD on projected Qt.X = U' . S. Vt
|
|
|
|
!
|
|
|
|
! 5. U = Q U'
|
|
|
|
END_DOC
|
|
|
|
|
|
|
|
integer, intent(in) :: LDA, LDU, LDVt, m, n, q, r
|
|
|
|
double precision, intent(in) :: A(LDA,n)
|
|
|
|
double precision, intent(out) :: U(LDU,r)
|
|
|
|
double precision,intent(out) :: Vt(LDVt,r)
|
|
|
|
double precision,intent(out) :: D(r)
|
|
|
|
integer :: i, j, k
|
|
|
|
|
|
|
|
double precision,allocatable :: Z(:,:), P(:,:), Y(:,:), UY(:,:)
|
|
|
|
double precision :: r1,r2
|
|
|
|
allocate(P(n,r), Z(m,r))
|
|
|
|
|
|
|
|
! P is a normal random matrix (n,r)
|
|
|
|
do k=1,r
|
|
|
|
do i=1,n
|
|
|
|
call random_number(r1)
|
|
|
|
call random_number(r2)
|
|
|
|
r1 = dsqrt(-2.d0*dlog(r1))
|
|
|
|
r2 = dtwo_pi*r2
|
|
|
|
P(i,k) = r1*dcos(r2)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
|
|
|
|
! Z(m,r) = A(m,n).P(n,r)
|
|
|
|
call dgemm('N','N',m,r,n,1.d0,A,size(A,1),P,size(P,1),0.d0,Z,size(Z,1))
|
|
|
|
|
|
|
|
! Power iterations
|
|
|
|
do k=1,q
|
|
|
|
! P(n,r) = At(n,m).Z(m,r)
|
|
|
|
call dgemm('T','N',n,r,m,1.d0,A,size(A,1),Z,size(Z,1),0.d0,P,size(P,1))
|
|
|
|
! Z(m,r) = A(m,n).P(n,r)
|
|
|
|
call dgemm('N','N',m,r,n,1.d0,A,size(A,1),P,size(P,1),0.d0,Z,size(Z,1))
|
|
|
|
enddo
|
|
|
|
|
|
|
|
deallocate(P)
|
|
|
|
|
|
|
|
! QR factorization of Z
|
|
|
|
call ortho_svd(Z,size(Z,1),m,r)
|
|
|
|
|
|
|
|
allocate(Y(r,n), UY(r,r))
|
|
|
|
! Y(r,n) = Zt(r,m).A(m,n)
|
|
|
|
call dgemm('T','N',r,n,m,1.d0,Z,size(Z,1),A,size(A,1),0.d0,Y,size(Y,1))
|
|
|
|
|
|
|
|
! SVD of Y
|
|
|
|
call svd(Y,size(Y,1),UY,size(UY,1),D,Vt,size(Vt,1),r,n)
|
|
|
|
deallocate(Y)
|
|
|
|
|
|
|
|
! U(m,r) = Z(m,r).UY(r,r)
|
|
|
|
call dgemm('N','N',m,r,r,1.d0,Z,size(Z,1),UY,size(UY,1),0.d0,U,size(U,1))
|
|
|
|
deallocate(UY,Z)
|
|
|
|
|
|
|
|
end
|
2019-01-25 11:39:31 +01:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
subroutine svd_complex(A,LDA,U,LDU,D,Vt,LDVt,m,n)
|
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Compute A = U.D.Vt
|
2020-05-25 11:31:28 +02:00
|
|
|
!
|
2019-12-02 19:25:35 +01:00
|
|
|
! LDx : leftmost dimension of x
|
|
|
|
!
|
|
|
|
! Dimension of A is m x n
|
|
|
|
! A,U,Vt are complex*16
|
|
|
|
! D is double precision
|
|
|
|
END_DOC
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
integer, intent(in) :: LDA, LDU, LDVt, m, n
|
|
|
|
complex*16, intent(in) :: A(LDA,n)
|
|
|
|
complex*16, intent(out) :: U(LDU,m)
|
|
|
|
complex*16, intent(out) :: Vt(LDVt,n)
|
|
|
|
double precision,intent(out) :: D(min(m,n))
|
|
|
|
complex*16,allocatable :: work(:)
|
|
|
|
double precision,allocatable :: rwork(:)
|
|
|
|
integer :: info, lwork, i, j, k, lrwork
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
complex*16,allocatable :: A_tmp(:,:)
|
2020-05-25 11:31:28 +02:00
|
|
|
allocate (A_tmp(LDA,n))
|
2019-12-02 19:25:35 +01:00
|
|
|
A_tmp = A
|
|
|
|
lrwork = 5*min(m,n)
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
! Find optimal size for temp arrays
|
|
|
|
allocate(work(1),rwork(lrwork))
|
|
|
|
lwork = -1
|
|
|
|
call zgesvd('A','A', m, n, A_tmp, LDA, &
|
|
|
|
D, U, LDU, Vt, LDVt, work, lwork, rwork, info)
|
|
|
|
lwork = int(work(1))
|
|
|
|
deallocate(work)
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
allocate(work(lwork))
|
|
|
|
call zgesvd('A','A', m, n, A_tmp, LDA, &
|
|
|
|
D, U, LDU, Vt, LDVt, work, lwork, rwork, info)
|
|
|
|
deallocate(work,rwork,A_tmp)
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
if (info /= 0) then
|
|
|
|
print *, info, ': SVD failed'
|
|
|
|
stop
|
|
|
|
endif
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
end
|
2020-05-25 11:31:28 +02:00
|
|
|
|
|
|
|
subroutine ortho_canonical_complex(overlap,LDA,N,C,LDC,m,cutoff)
|
2019-12-02 19:25:35 +01:00
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Compute C_new=C_old.U.s^-1/2 canonical orthogonalization.
|
|
|
|
!
|
2020-05-25 11:31:28 +02:00
|
|
|
! overlap : overlap matrix
|
2019-12-02 19:25:35 +01:00
|
|
|
!
|
|
|
|
! LDA : leftmost dimension of overlap array
|
|
|
|
!
|
|
|
|
! N : Overlap matrix is NxN (array is (LDA,N) )
|
|
|
|
!
|
|
|
|
! C : Coefficients of the vectors to orthogonalize. On exit,
|
|
|
|
! orthogonal vectors
|
|
|
|
!
|
|
|
|
! LDC : leftmost dimension of C
|
|
|
|
!
|
|
|
|
! m : Coefficients matrix is MxN, ( array is (LDC,N) )
|
|
|
|
!
|
|
|
|
END_DOC
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
integer, intent(in) :: lda, ldc, n
|
|
|
|
integer, intent(out) :: m
|
|
|
|
complex*16, intent(in) :: overlap(lda,n)
|
2020-05-25 11:31:28 +02:00
|
|
|
double precision, intent(in) :: cutoff
|
2019-12-02 19:25:35 +01:00
|
|
|
complex*16, intent(inout) :: C(ldc,n)
|
|
|
|
complex*16, allocatable :: U(:,:)
|
|
|
|
complex*16, allocatable :: Vt(:,:)
|
|
|
|
double precision, allocatable :: D(:)
|
|
|
|
complex*16, allocatable :: S(:,:)
|
|
|
|
!DIR$ ATTRIBUTES ALIGN : 64 :: U, Vt, D
|
|
|
|
integer :: info, i, j
|
2020-06-04 18:27:44 +02:00
|
|
|
double precision :: local_cutoff
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
if (n < 2) then
|
|
|
|
return
|
|
|
|
endif
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
allocate (U(ldc,n), Vt(lda,n), D(n), S(lda,n))
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
call svd_complex(overlap,lda,U,ldc,D,Vt,lda,n,n)
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
D(:) = dsqrt(D(:))
|
2020-06-04 18:27:44 +02:00
|
|
|
local_cutoff = dsqrt(cutoff)*D(1) ! such that D(i)/D(1) > dsqrt(cutoff) is kept
|
2019-12-02 19:25:35 +01:00
|
|
|
m=n
|
|
|
|
do i=1,n
|
2020-06-04 18:27:44 +02:00
|
|
|
if ( D(i) >= local_cutoff ) then
|
2019-12-02 19:25:35 +01:00
|
|
|
D(i) = 1.d0/D(i)
|
|
|
|
else
|
|
|
|
m = i-1
|
2020-06-04 18:27:44 +02:00
|
|
|
print *, 'Removed Linear dependencies below:', local_cutoff
|
2019-12-02 19:25:35 +01:00
|
|
|
exit
|
|
|
|
endif
|
|
|
|
enddo
|
2020-05-25 11:31:28 +02:00
|
|
|
do i=m+1,n
|
2019-12-02 19:25:35 +01:00
|
|
|
D(i) = 0.d0
|
2020-05-25 11:31:28 +02:00
|
|
|
enddo
|
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
do j=1,n
|
|
|
|
do i=1,n
|
|
|
|
S(i,j) = U(i,j)*D(j)
|
|
|
|
enddo
|
|
|
|
enddo
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
do j=1,n
|
|
|
|
do i=1,n
|
|
|
|
U(i,j) = C(i,j)
|
|
|
|
enddo
|
|
|
|
enddo
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
call zgemm('N','N',n,n,n,(1.d0,0.d0),U,size(U,1),S,size(S,1),(0.d0,0.d0),C,size(C,1))
|
2020-05-25 11:31:28 +02:00
|
|
|
deallocate (U, Vt, D, S)
|
|
|
|
|
|
|
|
end
|
|
|
|
|
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
subroutine ortho_qr_complex(A,LDA,m,n)
|
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Orthogonalization using Q.R factorization
|
2020-05-25 11:31:28 +02:00
|
|
|
!
|
2019-12-02 19:25:35 +01:00
|
|
|
! A : matrix to orthogonalize
|
|
|
|
!
|
|
|
|
! LDA : leftmost dimension of A
|
|
|
|
!
|
|
|
|
! n : Number of rows of A
|
|
|
|
!
|
|
|
|
! m : Number of columns of A
|
|
|
|
!
|
|
|
|
END_DOC
|
|
|
|
integer, intent(in) :: m,n, LDA
|
|
|
|
complex*16, intent(inout) :: A(LDA,n)
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
integer :: lwork, info
|
|
|
|
integer, allocatable :: jpvt(:)
|
|
|
|
complex*16, allocatable :: tau(:), work(:)
|
|
|
|
|
|
|
|
allocate (jpvt(n), tau(n), work(1))
|
|
|
|
LWORK=-1
|
|
|
|
call zgeqrf( m, n, A, LDA, TAU, WORK, LWORK, INFO )
|
|
|
|
LWORK=2*int(WORK(1))
|
|
|
|
deallocate(WORK)
|
|
|
|
allocate(WORK(LWORK))
|
|
|
|
call zgeqrf(m, n, A, LDA, TAU, WORK, LWORK, INFO )
|
|
|
|
call zungqr(m, n, n, A, LDA, tau, WORK, LWORK, INFO)
|
|
|
|
deallocate(WORK,jpvt,tau)
|
|
|
|
end
|
|
|
|
|
|
|
|
subroutine ortho_qr_unblocked_complex(A,LDA,m,n)
|
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Orthogonalization using Q.R factorization
|
|
|
|
!
|
|
|
|
! A : matrix to orthogonalize
|
|
|
|
!
|
|
|
|
! LDA : leftmost dimension of A
|
|
|
|
!
|
|
|
|
! n : Number of rows of A
|
|
|
|
!
|
|
|
|
! m : Number of columns of A
|
|
|
|
!
|
|
|
|
END_DOC
|
|
|
|
integer, intent(in) :: m,n, LDA
|
|
|
|
double precision, intent(inout) :: A(LDA,n)
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
integer :: info
|
|
|
|
integer, allocatable :: jpvt(:)
|
|
|
|
double precision, allocatable :: tau(:), work(:)
|
|
|
|
|
|
|
|
print *, irp_here, ': TO DO'
|
|
|
|
stop -1
|
|
|
|
|
|
|
|
! allocate (jpvt(n), tau(n), work(n))
|
|
|
|
! call dgeqr2( m, n, A, LDA, TAU, WORK, INFO )
|
|
|
|
! call dorg2r(m, n, n, A, LDA, tau, WORK, INFO)
|
|
|
|
! deallocate(WORK,jpvt,tau)
|
|
|
|
end
|
2020-05-25 11:31:28 +02:00
|
|
|
|
|
|
|
subroutine ortho_lowdin_complex(overlap,LDA,N,C,LDC,m,cutoff)
|
2019-12-02 19:25:35 +01:00
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Compute C_new=C_old.S^-1/2 orthogonalization.
|
|
|
|
!
|
2020-05-25 11:31:28 +02:00
|
|
|
! overlap : overlap matrix
|
2019-12-02 19:25:35 +01:00
|
|
|
!
|
|
|
|
! LDA : leftmost dimension of overlap array
|
|
|
|
!
|
|
|
|
! N : Overlap matrix is NxN (array is (LDA,N) )
|
|
|
|
!
|
|
|
|
! C : Coefficients of the vectors to orthogonalize. On exit,
|
|
|
|
! orthogonal vectors
|
|
|
|
!
|
|
|
|
! LDC : leftmost dimension of C
|
|
|
|
!
|
|
|
|
! M : Coefficients matrix is MxN, ( array is (LDC,N) )
|
|
|
|
!
|
|
|
|
END_DOC
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
integer, intent(in) :: LDA, ldc, n, m
|
|
|
|
complex*16, intent(in) :: overlap(lda,n)
|
|
|
|
complex*16, intent(inout) :: C(ldc,n)
|
|
|
|
complex*16, allocatable :: U(:,:)
|
|
|
|
complex*16, allocatable :: Vt(:,:)
|
|
|
|
double precision, allocatable :: D(:)
|
|
|
|
complex*16, allocatable :: S(:,:)
|
2020-05-25 11:31:28 +02:00
|
|
|
double precision, intent(in) :: cutoff
|
2020-06-04 18:27:44 +02:00
|
|
|
double precision :: local_cutoff
|
|
|
|
integer :: info, i, j, k, mm
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
if (n < 2) then
|
|
|
|
return
|
|
|
|
endif
|
|
|
|
|
|
|
|
allocate(U(ldc,n),Vt(lda,n),S(lda,n),D(n))
|
|
|
|
|
|
|
|
call svd_complex(overlap,lda,U,ldc,D,Vt,lda,n,n)
|
2020-06-04 18:27:44 +02:00
|
|
|
D(:) = dsqrt(D(:))
|
|
|
|
local_cutoff = dsqrt(cutoff)*D(1) ! such that D(i)/D(1) > dsqrt(cutoff) is kept
|
|
|
|
mm=n
|
2019-12-02 19:25:35 +01:00
|
|
|
do i=1,n
|
2020-06-04 18:27:44 +02:00
|
|
|
if ( D(i) >= local_cutoff) then
|
|
|
|
D(i) = 1.d0/D(i)
|
2019-12-02 19:25:35 +01:00
|
|
|
else
|
2020-06-04 18:27:44 +02:00
|
|
|
mm = mm-1
|
|
|
|
D(i) = 0.d0
|
2019-12-02 19:25:35 +01:00
|
|
|
endif
|
|
|
|
do j=1,n
|
|
|
|
S(j,i) = (0.d0,0.d0)
|
|
|
|
enddo
|
|
|
|
enddo
|
2020-06-04 18:27:44 +02:00
|
|
|
|
|
|
|
if (mm < n) then
|
|
|
|
print *, 'Removed Linear dependencies below ', local_cutoff
|
|
|
|
endif
|
|
|
|
|
|
|
|
!$OMP PARALLEL DEFAULT(NONE) &
|
|
|
|
!$OMP SHARED(S,U,D,Vt,n,C,m,local_cutoff) &
|
|
|
|
!$OMP PRIVATE(i,j,k)
|
2019-12-02 19:25:35 +01:00
|
|
|
|
|
|
|
do k=1,n
|
|
|
|
if (D(k) /= 0.d0) then
|
2020-06-04 18:27:44 +02:00
|
|
|
!$OMP DO SCHEDULE(STATIC)
|
2019-12-02 19:25:35 +01:00
|
|
|
do j=1,n
|
|
|
|
do i=1,n
|
|
|
|
S(i,j) = S(i,j) + U(i,k)*D(k)*Vt(k,j)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
!$OMP END DO NOWAIT
|
|
|
|
endif
|
|
|
|
enddo
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
!$OMP BARRIER
|
|
|
|
!$OMP DO
|
|
|
|
do j=1,n
|
|
|
|
do i=1,m
|
|
|
|
U(i,j) = C(i,j)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
!$OMP END DO
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
!$OMP END PARALLEL
|
|
|
|
|
|
|
|
call zgemm('N','N',m,n,n,(1.d0,0.d0),U,size(U,1),S,size(S,1),(0.d0,0.d0),C,size(C,1))
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
deallocate(U,Vt,S,D)
|
|
|
|
end
|
|
|
|
|
|
|
|
subroutine get_inverse_complex(A,LDA,m,C,LDC)
|
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Returns the inverse of the square matrix A
|
|
|
|
END_DOC
|
|
|
|
integer, intent(in) :: m, LDA, LDC
|
|
|
|
complex*16, intent(in) :: A(LDA,m)
|
|
|
|
complex*16, intent(out) :: C(LDC,m)
|
|
|
|
|
|
|
|
integer :: info,lwork
|
|
|
|
integer, allocatable :: ipiv(:)
|
|
|
|
complex*16,allocatable :: work(:)
|
|
|
|
allocate (ipiv(m), work(m*m))
|
|
|
|
lwork = size(work)
|
|
|
|
C(1:m,1:m) = A(1:m,1:m)
|
|
|
|
call zgetrf(m,m,C,size(C,1),ipiv,info)
|
|
|
|
if (info /= 0) then
|
|
|
|
print *, info
|
|
|
|
stop 'error in inverse (zgetrf)'
|
|
|
|
endif
|
|
|
|
call zgetri(m,C,size(C,1),ipiv,work,lwork,info)
|
|
|
|
if (info /= 0) then
|
|
|
|
print *, info
|
|
|
|
stop 'error in inverse (zgetri)'
|
|
|
|
endif
|
|
|
|
deallocate(ipiv,work)
|
|
|
|
end
|
|
|
|
|
|
|
|
|
2020-05-25 11:31:28 +02:00
|
|
|
subroutine get_pseudo_inverse_complex(A,LDA,m,n,C,LDC,cutoff)
|
2019-12-02 19:25:35 +01:00
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Find C = A^-1
|
|
|
|
END_DOC
|
|
|
|
integer, intent(in) :: m,n, LDA, LDC
|
|
|
|
complex*16, intent(in) :: A(LDA,n)
|
2024-02-08 14:13:10 +01:00
|
|
|
double precision, intent(in) :: cutoff
|
2019-12-02 19:25:35 +01:00
|
|
|
complex*16, intent(out) :: C(LDC,m)
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
double precision, allocatable :: D(:), rwork(:)
|
|
|
|
complex*16, allocatable :: U(:,:), Vt(:,:), work(:), A_tmp(:,:)
|
|
|
|
integer :: info, lwork
|
|
|
|
integer :: i,j,k
|
2024-02-08 14:13:10 +01:00
|
|
|
double precision :: d1
|
2019-12-02 19:25:35 +01:00
|
|
|
allocate (D(n),U(m,n),Vt(n,n),work(1),A_tmp(m,n),rwork(5*n))
|
|
|
|
do j=1,n
|
|
|
|
do i=1,m
|
|
|
|
A_tmp(i,j) = A(i,j)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
lwork = -1
|
|
|
|
call zgesvd('S','A', m, n, A_tmp, m,D,U,m,Vt,n,work,lwork,rwork,info)
|
|
|
|
if (info /= 0) then
|
|
|
|
print *, info, ': SVD failed'
|
|
|
|
stop
|
|
|
|
endif
|
|
|
|
lwork = int(real(work(1)))
|
|
|
|
deallocate(work)
|
|
|
|
allocate(work(lwork))
|
|
|
|
call zgesvd('S','A', m, n, A_tmp, m,D,U,m,Vt,n,work,lwork,rwork,info)
|
|
|
|
if (info /= 0) then
|
|
|
|
print *, info, ':: SVD failed'
|
|
|
|
stop 1
|
|
|
|
endif
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2024-01-26 13:19:21 +01:00
|
|
|
d1 = D(1)
|
2019-12-02 19:25:35 +01:00
|
|
|
do i=1,n
|
2024-01-26 13:19:21 +01:00
|
|
|
if (D(i) > cutoff*d1) then
|
2019-12-02 19:25:35 +01:00
|
|
|
D(i) = 1.d0/D(i)
|
|
|
|
else
|
|
|
|
D(i) = 0.d0
|
|
|
|
endif
|
|
|
|
enddo
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
C = (0.d0,0.d0)
|
|
|
|
do i=1,m
|
|
|
|
do j=1,n
|
|
|
|
do k=1,n
|
|
|
|
C(j,i) = C(j,i) + U(i,k) * D(k) * Vt(k,j)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
enddo
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
deallocate(U,D,Vt,work,A_tmp,rwork)
|
2020-05-25 11:31:28 +02:00
|
|
|
|
2019-12-02 19:25:35 +01:00
|
|
|
end
|
|
|
|
|
|
|
|
subroutine lapack_diagd_diag_in_place_complex(eigvalues,eigvectors,nmax,n)
|
|
|
|
implicit none
|
|
|
|
BEGIN_DOC
|
|
|
|
! Diagonalize matrix H(complex)
|
|
|
|
!
|
|
|
|
! H is untouched between input and ouptut
|
|
|
|
!
|
|
|
|
! eigevalues(i) = ith lowest eigenvalue of the H matrix
|
|
|
|
!
|
|
|
|
! eigvectors(i,j) = <i|psi_j> where i is the basis function and psi_j is the j th eigenvector
|
|
|
|
!
|
|
|
|
END_DOC
|
|
|
|
integer, intent(in) :: n,nmax
|
|
|
|
! double precision, intent(out) :: eigvectors(nmax,n)
|
|
|
|
complex*16, intent(inout) :: eigvectors(nmax,n)
|
|
|
|
double precision, intent(out) :: eigvalues(n)
|
|
|
|
! double precision, intent(in) :: H(nmax,n)
|
|
|
|
complex*16,allocatable :: work(:)
|
|
|
|
integer ,allocatable :: iwork(:)
|
|
|
|
! complex*16,allocatable :: A(:,:)
|
|
|
|
double precision, allocatable :: rwork(:)
|
|
|
|
integer :: lrwork, lwork, info, i,j,l,k, liwork
|
|
|
|
|
|
|
|
! print*,'Diagonalization by jacobi'
|
|
|
|
! print*,'n = ',n
|
|
|
|
|
|
|
|
lwork = 2*n*n + 2*n
|
|
|
|
lrwork = 2*n*n + 5*n+ 1
|
|
|
|
liwork = 5*n + 3
|
|
|
|
allocate (work(lwork),iwork(liwork),rwork(lrwork))
|
|
|
|
|
|
|
|
lwork = -1
|
|
|
|
liwork = -1
|
|
|
|
lrwork = -1
|
|
|
|
! get optimal work size
|
|
|
|
call ZHEEVD( 'V', 'U', n, eigvectors, nmax, eigvalues, work, lwork, &
|
|
|
|
rwork, lrwork, iwork, liwork, info )
|
|
|
|
if (info < 0) then
|
|
|
|
print *, irp_here, ': ZHEEVD: the ',-info,'-th argument had an illegal value'
|
|
|
|
stop 2
|
|
|
|
endif
|
|
|
|
lwork = int( real(work(1)))
|
|
|
|
liwork = iwork(1)
|
|
|
|
lrwork = int(rwork(1))
|
|
|
|
deallocate (work,iwork,rwork)
|
|
|
|
|
|
|
|
allocate (work(lwork),iwork(liwork),rwork(lrwork))
|
|
|
|
call ZHEEVD( 'V', 'U', n, eigvectors, nmax, eigvalues, work, lwork, &
|
|
|
|
rwork, lrwork, iwork, liwork, info )
|
|
|
|
deallocate(work,iwork,rwork)
|
|
|
|
|
|
|
|
|
|
|
|
if (info < 0) then
|
|
|
|
print *, irp_here, ': ZHEEVD: the ',-info,'-th argument had an illegal value'
|
|
|
|
stop 2
|
|
|
|
else if( info > 0 ) then
|
|
|
|
write(*,*)'ZHEEVD Failed; calling ZHEEV'
|
|
|
|
lwork = 2*n - 1
|
|
|
|
lrwork = 3*n - 2
|
|
|
|
allocate(work(lwork),rwork(lrwork))
|
|
|
|
lwork = -1
|
|
|
|
call ZHEEV('V','L',n,eigvectors,nmax,eigvalues,work,lwork,rwork,info)
|
|
|
|
if (info < 0) then
|
|
|
|
print *, irp_here, ': ZHEEV: the ',-info,'-th argument had an illegal value'
|
|
|
|
stop 2
|
|
|
|
endif
|
|
|
|
lwork = int(work(1))
|
|
|
|
deallocate(work)
|
|
|
|
allocate(work(lwork))
|
|
|
|
call ZHEEV('V','L',n,eigvectors,nmax,eigvalues,work,lwork,rwork,info)
|
|
|
|
if (info /= 0 ) then
|
|
|
|
write(*,*)'ZHEEV Failed'
|
|
|
|
stop 1
|
|
|
|
endif
|
|
|
|
deallocate(work,rwork)
|
|
|
|
end if
|
|
|
|
|
|
|
|
end
|
|
|
|
|
|
|
|
subroutine lapack_diagd_diag_complex(eigvalues,eigvectors,H,nmax,n)
|
2020-05-25 11:31:28 +02:00
|
|
|
implicit none
|
2019-12-02 19:25:35 +01:00
|
|
|
BEGIN_DOC
|
|
|
|
! Diagonalize matrix H(complex)
|
|
|
|
!
|
|
|
|
! H is untouched between input and ouptut
|
|
|
|
!
|
|
|
|
! eigevalues(i) = ith lowest eigenvalue of the H matrix
|
|
|
|
!
|
|
|
|
! eigvectors(i,j) = <i|psi_j> where i is the basis function and psi_j is the j th eigenvector
|
|
|
|
!
|
|
|
|
END_DOC
|
|
|
|
integer, intent(in) :: n,nmax
|
|
|
|
! double precision, intent(out) :: eigvectors(nmax,n)
|
|
|
|
complex*16, intent(out) :: eigvectors(nmax,n)
|
|
|
|
double precision, intent(out) :: eigvalues(n)
|
|
|
|
! double precision, intent(in) :: H(nmax,n)
|
|
|
|
complex*16, intent(in) :: H(nmax,n)
|
|
|
|
double precision, allocatable :: eigenvalues(:)
|
|
|
|
complex*16,allocatable :: work(:)
|
|
|
|
integer ,allocatable :: iwork(:)
|
|
|
|
complex*16,allocatable :: A(:,:)
|
|
|
|
double precision, allocatable :: rwork(:)
|
|
|
|
integer :: lrwork, lwork, info, i,j,l,k, liwork
|
|
|
|
|
|
|
|
allocate(A(nmax,n),eigenvalues(n))
|
|
|
|
! print*,'Diagonalization by jacobi'
|
|
|
|
! print*,'n = ',n
|
|
|
|
|
|
|
|
A=H
|
|
|
|
lwork = 2*n*n + 2*n
|
|
|
|
lrwork = 2*n*n + 5*n+ 1
|
|
|
|
liwork = 5*n + 3
|
|
|
|
allocate (work(lwork),iwork(liwork),rwork(lrwork))
|
|
|
|
|
|
|
|
lwork = -1
|
|
|
|
liwork = -1
|
|
|
|
lrwork = -1
|
|
|
|
! get optimal work size
|
|
|
|
call ZHEEVD( 'V', 'U', n, A, nmax, eigenvalues, work, lwork, &
|
|
|
|
rwork, lrwork, iwork, liwork, info )
|
|
|
|
if (info < 0) then
|
|
|
|
print *, irp_here, ': ZHEEVD: the ',-info,'-th argument had an illegal value'
|
|
|
|
stop 2
|
|
|
|
endif
|
|
|
|
lwork = int( real(work(1)))
|
|
|
|
liwork = iwork(1)
|
|
|
|
lrwork = int(rwork(1))
|
|
|
|
deallocate (work,iwork,rwork)
|
|
|
|
|
|
|
|
allocate (work(lwork),iwork(liwork),rwork(lrwork))
|
|
|
|
call ZHEEVD( 'V', 'U', n, A, nmax, eigenvalues, work, lwork, &
|
|
|
|
rwork, lrwork, iwork, liwork, info )
|
|
|
|
deallocate(work,iwork,rwork)
|
|
|
|
|
|
|
|
if (info < 0) then
|
|
|
|
print *, irp_here, ': ZHEEVD: the ',-info,'-th argument had an illegal value'
|
|
|
|
stop 2
|
|
|
|
else if( info > 0 ) then
|
|
|
|
write(*,*)'ZHEEVD Failed; calling ZHEEV'
|
|
|
|
lwork = 2*n - 1
|
|
|
|
lrwork = 3*n - 2
|
|
|
|
allocate(work(lwork),rwork(lrwork))
|
|
|
|
lwork = -1
|
|
|
|
call ZHEEV('V','L',n,A,nmax,eigenvalues,work,lwork,rwork,info)
|
|
|
|
if (info < 0) then
|
|
|
|
print *, irp_here, ': ZHEEV: the ',-info,'-th argument had an illegal value'
|
|
|
|
stop 2
|
|
|
|
endif
|
|
|
|
lwork = int(work(1))
|
|
|
|
deallocate(work)
|
|
|
|
allocate(work(lwork))
|
|
|
|
call ZHEEV('V','L',n,A,nmax,eigenvalues,work,lwork,rwork,info)
|
|
|
|
if (info /= 0 ) then
|
|
|
|
write(*,*)'ZHEEV Failed'
|
|
|
|
stop 1
|
|
|
|
endif
|
|
|
|
deallocate(work,rwork)
|
|
|
|
end if
|
|
|
|
|
|
|
|
eigvectors = (0.d0,0.d0)
|
|
|
|
eigvalues = 0.d0
|
|
|
|
do j = 1, n
|
|
|
|
eigvalues(j) = eigenvalues(j)
|
|
|
|
do i = 1, n
|
|
|
|
eigvectors(i,j) = A(i,j)
|
|
|
|
enddo
|
|
|
|
enddo
|
|
|
|
deallocate(A,eigenvalues)
|
|
|
|
end
|
|
|
|
|
|
|
|
subroutine lapack_diagd_complex(eigvalues,eigvectors,H,nmax,n)
|
|
|
|