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Added randomized SVD routines
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@ -11,7 +11,7 @@ subroutine svd(A,LDA,U,LDU,D,Vt,LDVt,m,n)
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integer, intent(in) :: LDA, LDU, LDVt, m, n
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double precision, intent(in) :: A(LDA,n)
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double precision, intent(out) :: U(LDU,m)
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double precision, intent(out) :: U(LDU,min(m,n))
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double precision,intent(out) :: Vt(LDVt,n)
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double precision,intent(out) :: D(min(m,n))
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double precision,allocatable :: work(:)
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@ -24,14 +24,14 @@ subroutine svd(A,LDA,U,LDU,D,Vt,LDVt,m,n)
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! Find optimal size for temp arrays
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allocate(work(1))
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lwork = -1
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call dgesvd('A','A', m, n, A_tmp, LDA, &
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call dgesvd('S','S', m, n, A_tmp, LDA, &
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D, U, LDU, Vt, LDVt, work, lwork, info)
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! /!\ int(WORK(1)) becomes negative when WORK(1) > 2147483648
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lwork = max(int(work(1)), 5*MIN(M,N))
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deallocate(work)
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allocate(work(lwork))
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call dgesvd('A','A', m, n, A_tmp, LDA, &
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call dgesvd('S','S', m, n, A_tmp, LDA, &
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D, U, LDU, Vt, LDVt, work, lwork, info)
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deallocate(work,A_tmp)
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@ -42,6 +42,128 @@ subroutine svd(A,LDA,U,LDU,D,Vt,LDVt,m,n)
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end
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subroutine eigSVD(A,LDA,U,LDU,D,Vt,LDVt,m,n)
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implicit none
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BEGIN_DOC
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! Algorithm 3 of https://arxiv.org/pdf/1810.06860.pdf
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!
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! A(m,n) = U(m,n) D(n) Vt(n,n) with m>n
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END_DOC
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integer, intent(in) :: LDA, LDU, LDVt, m, n
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double precision, intent(in) :: A(LDA,n)
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double precision, intent(out) :: U(LDU,n)
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double precision,intent(out) :: Vt(LDVt,n)
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double precision,intent(out) :: D(n)
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integer :: i,j,k
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if (m<n) then
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stop -1
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call svd(A,LDA,U,LDU,D,Vt,LDVt,m,n)
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return
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endif
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double precision, allocatable :: B(:,:), V(:,:)
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allocate(B(n,n))
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! B = - At . A
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call dgemm('T','N',n,n,m,-1.d0,A,size(A,1),A,size(A,1),0.d0,B,size(B,1))
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! V, D = eig(B)
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allocate(V(n,n))
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call lapack_diagd(D,V,B,n,n)
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deallocate(B)
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do j=1,n
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do i=1,n
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Vt(i,j) = V(j,i)
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enddo
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enddo
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! S = sqrt(-D)
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! U = A.V.S^-1
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! U = A.(S^-1.vt)t
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do k=1,n
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if (D(k) >= 0.d0) then
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exit
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endif
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D(k) = dsqrt(-D(k))
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call dscal(n, 1.d0/D(k), V(1,k), 1)
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enddo
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D(k:n) = 0.d0
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k=k-1
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call dgemm('N','N',m,n,k,1.d0,A,size(A,1),V,size(V,1),0.d0,U,size(U,1))
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end
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subroutine randomized_svd(A,LDA,U,LDU,D,Vt,LDVt,m,n,q,r)
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implicit none
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include 'constants.include.F'
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BEGIN_DOC
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! Randomized SVD: rank r, q power iterations
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!
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! 1. Sample column space of A with P: Z = A.P where P is random with r+p columns.
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!
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! 2. Power iterations : Z <- X . (Xt.Z)
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!
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! 3. Z = Q.R
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!
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! 4. Compute SVD on projected Qt.X = U' . S. Vt
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!
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! 5. U = Q U'
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END_DOC
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integer, intent(in) :: LDA, LDU, LDVt, m, n, q, r
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double precision, intent(in) :: A(LDA,n)
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double precision, intent(out) :: U(LDU,r)
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double precision,intent(out) :: Vt(LDVt,r)
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double precision,intent(out) :: D(r)
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integer :: i, j, k
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double precision,allocatable :: Z(:,:), P(:,:), Y(:,:), UY(:,:)
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double precision :: r1,r2
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allocate(P(n,r), Z(m,r))
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! P is a normal random matrix (n,r)
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do k=1,r
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do i=1,n
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call random_number(r1)
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call random_number(r2)
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r1 = dsqrt(-2.d0*dlog(r1))
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r2 = dtwo_pi*r2
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P(i,k) = r1*dcos(r2)
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enddo
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enddo
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! Z(m,r) = A(m,n).P(n,r)
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call dgemm('N','N',m,r,n,1.d0,A,size(A,1),P,size(P,1),0.d0,Z,size(Z,1))
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! Power iterations
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do k=1,q
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! P(n,r) = At(n,m).Z(m,r)
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call dgemm('T','N',n,r,m,1.d0,A,size(A,1),Z,size(Z,1),0.d0,P,size(P,1))
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! Z(m,r) = A(m,n).P(n,r)
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call dgemm('N','N',m,r,n,1.d0,A,size(A,1),P,size(P,1),0.d0,Z,size(Z,1))
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enddo
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deallocate(P)
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! QR factorization of Z
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call ortho_svd(Z,size(Z,1),m,r)
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allocate(Y(r,n), UY(r,r))
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! Y(r,n) = Zt(r,m).A(m,n)
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call dgemm('T','N',r,n,m,1.d0,Z,size(Z,1),A,size(A,1),0.d0,Y,size(Y,1))
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! SVD of Y
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call svd(Y,size(Y,1),UY,size(UY,1),D,Vt,size(Vt,1),r,n)
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deallocate(Y)
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! U(m,r) = Z(m,r).UY(r,r)
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call dgemm('N','N',m,r,r,1.d0,Z,size(Z,1),UY,size(UY,1),0.d0,U,size(U,1))
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deallocate(UY,Z)
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end
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subroutine svd_complex(A,LDA,U,LDU,D,Vt,LDVt,m,n)
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implicit none
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@ -807,6 +929,33 @@ subroutine ortho_canonical(overlap,LDA,N,C,LDC,m,cutoff)
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end
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subroutine ortho_svd(A,LDA,m,n)
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implicit none
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BEGIN_DOC
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! Orthogonalization via fast SVD
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!
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! A : matrix to orthogonalize
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!
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! LDA : leftmost dimension of A
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!
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! m : Number of rows of A
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!
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! n : Number of columns of A
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!
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END_DOC
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integer, intent(in) :: m,n, LDA
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double precision, intent(inout) :: A(LDA,n)
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if (m < n) then
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call ortho_qr(A,LDA,m,n)
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endif
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double precision, allocatable :: U(:,:), D(:), Vt(:,:)
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allocate(U(m,n), D(n), Vt(n,n))
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call SVD(A,LDA,U,size(U,1),D,Vt,size(Vt,1),m,n)
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A(1:m,1:n) = U(1:m,1:n)
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deallocate(U,D, Vt)
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end
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subroutine ortho_qr(A,LDA,m,n)
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implicit none
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BEGIN_DOC
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