mirror of
https://github.com/QuantumPackage/qp2.git
synced 2024-12-22 11:33:29 +01:00
326 lines
9.2 KiB
Fortran
326 lines
9.2 KiB
Fortran
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subroutine get_inv_half_svd(matrix, n, matrix_inv_half)
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BEGIN_DOC
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! :math:`X = S^{-1/2}` obtained by SVD
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END_DOC
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implicit none
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integer, intent(in) :: n
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double precision, intent(in) :: matrix(n,n)
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double precision, intent(out) :: matrix_inv_half(n,n)
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integer :: num_linear_dependencies
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integer :: LDA, LDC
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integer :: info, i, j, k
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double precision, parameter :: threshold = 1.d-6
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double precision, allocatable :: U(:,:),Vt(:,:), D(:),matrix_half(:,:),D_half(:)
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double precision :: accu_d,accu_nd
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LDA = size(matrix, 1)
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LDC = size(matrix_inv_half, 1)
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if(LDA .ne. LDC) then
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print*, ' LDA != LDC'
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stop
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endif
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print*, ' n = ', n
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print*, ' LDA = ', LDA
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print*, ' LDC = ', LDC
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double precision,allocatable :: WR(:),WI(:),VL(:,:),VR(:,:)
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allocate(WR(n),WI(n),VL(n,n),VR(n,n))
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call lapack_diag_non_sym(n,matrix,WR,WI,VL,VR)
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do i = 1, n
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print*,'WR,WI',WR(i),WI(i)
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enddo
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allocate(U(LDC,n), Vt(LDA,n), D(n))
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call svd(matrix, LDA, U, LDC, D, Vt, LDA, n, n)
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double precision, allocatable :: tmp1(:,:),tmp2(:,:),D_mat(:,:)
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allocate(tmp1(n,n),tmp2(n,n),D_mat(n,n),matrix_half(n,n),D_half(n))
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D_mat = 0.d0
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do i = 1,n
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D_mat(i,i) = D(i)
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enddo
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! matrix = U D Vt
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! tmp1 = U D
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tmp1 = 0.d0
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call dgemm( 'N', 'N', n, n, n, 1.d0 &
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, U, size(U, 1), D_mat, size(D_mat, 1) &
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, 0.d0, tmp1, size(tmp1, 1) )
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! tmp2 = tmp1 X Vt = matrix
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tmp2 = 0.d0
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call dgemm( 'N', 'N', n, n, n, 1.d0 &
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, tmp1, size(tmp1, 1), Vt, size(Vt, 1) &
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, 0.d0, tmp2, size(tmp2, 1) )
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print*,'Checking the recomposition of the matrix'
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accu_nd = 0.d0
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accu_d = 0.d0
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do i = 1, n
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accu_d += dabs(tmp2(i,i) - matrix(i,i))
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do j = 1, n
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if(i==j)cycle
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accu_nd += dabs(tmp2(j,i) - matrix(j,i))
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enddo
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enddo
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print*,'accu_d =',accu_d
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print*,'accu_nd =',accu_nd
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print*,'passed the recomposition'
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num_linear_dependencies = 0
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do i = 1, n
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if(abs(D(i)) <= threshold) then
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D(i) = 0.d0
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num_linear_dependencies += 1
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else
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ASSERT (D(i) > 0.d0)
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D_half(i) = dsqrt(D(i))
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D(i) = 1.d0 / dsqrt(D(i))
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endif
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enddo
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write(*,*) ' linear dependencies', num_linear_dependencies
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matrix_inv_half = 0.d0
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matrix_half = 0.d0
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do k = 1, n
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if(D(k) /= 0.d0) then
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do j = 1, n
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do i = 1, n
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! matrix_inv_half(i,j) = matrix_inv_half(i,j) + U(i,k) * D(k) * Vt(k,j)
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matrix_inv_half(i,j) = matrix_inv_half(i,j) + U(i,k) * D(k) * Vt(j,k)
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matrix_half(i,j) = matrix_half(i,j) + U(i,k) * D_half(k) * Vt(j,k)
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enddo
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enddo
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endif
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enddo
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print*,'testing S^1/2 * S^1/2= S'
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! tmp1 = S^1/2 X S^1/2
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tmp1 = 0.d0
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call dgemm( 'N', 'N', n, n, n, 1.d0 &
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, matrix_half, size(matrix_half, 1), matrix_half, size(matrix_half, 1) &
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, 0.d0, tmp1, size(tmp1, 1) )
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accu_nd = 0.d0
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accu_d = 0.d0
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do i = 1, n
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accu_d += dabs(tmp1(i,i) - matrix(i,i))
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do j = 1, n
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if(i==j)cycle
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accu_nd += dabs(tmp1(j,i) - matrix(j,i))
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enddo
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enddo
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print*,'accu_d =',accu_d
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print*,'accu_nd =',accu_nd
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! print*,'S inv half'
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! do i = 1, n
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! write(*, '(1000(F16.10,X))') matrix_inv_half(i,:)
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! enddo
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double precision, allocatable :: pseudo_inverse(:,:),identity(:,:)
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allocate( pseudo_inverse(n,n),identity(n,n))
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call get_pseudo_inverse(matrix,n,n,n,pseudo_inverse,n,threshold)
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! S^-1 X S = 1
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! identity = 0.d0
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! call dgemm( 'N', 'N', n, n, n, 1.d0 &
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! , matrix, size(matrix, 1), pseudo_inverse, size(pseudo_inverse, 1) &
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! , 0.d0, identity, size(identity, 1) )
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print*,'Checking S^-1/2 X S^-1/2 = S^-1 ?'
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! S^-1/2 X S^-1/2 = S^-1 ?
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tmp1 = 0.d0
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call dgemm( 'N', 'N', n, n, n, 1.d0 &
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,matrix_inv_half, size(matrix_inv_half, 1), matrix_inv_half, size(matrix_inv_half, 1) &
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, 0.d0, tmp1, size(tmp1, 1) )
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accu_nd = 0.d0
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accu_d = 0.d0
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do i = 1, n
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accu_d += dabs(1.d0 - pseudo_inverse(i,i))
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do j = 1, n
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if(i==j)cycle
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accu_nd += dabs(tmp1(j,i) - pseudo_inverse(j,i))
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enddo
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enddo
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print*,'accu_d =',accu_d
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print*,'accu_nd =',accu_nd
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stop
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!
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! ! ( S^-1/2 x S ) x S^-1/2
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! Stmp2 = 0.d0
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! call dgemm( 'N', 'N', n, n, n, 1.d0 &
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! , Stmp, size(Stmp, 1), matrix_inv_half, size(matrix_inv_half, 1) &
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! , 0.d0, Stmp2, size(Stmp2, 1) )
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! S^-1/2 x ( S^-1/2 x S )
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! Stmp2 = 0.d0
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! call dgemm( 'N', 'N', n, n, n, 1.d0 &
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! , matrix_inv_half, size(matrix_inv_half, 1), Stmp, size(Stmp, 1) &
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! , 0.d0, Stmp2, size(Stmp2, 1) )
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! do i = 1, n
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! do j = 1, n
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! if(i==j) then
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! accu_d += Stmp2(j,i)
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! else
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! accu_nd = accu_nd + Stmp2(j,i) * Stmp2(j,i)
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! endif
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! enddo
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! enddo
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! accu_nd = dsqrt(accu_nd)
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! print*, ' after S^-1/2: sum of off-diag S elements = ', accu_nd
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! print*, ' after S^-1/2: sum of diag S elements = ', accu_d
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! do i = 1, n
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! write(*,'(1000(F16.10,X))') Stmp2(i,:)
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! enddo
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!double precision :: thresh
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!thresh = 1.d-10
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!if( accu_nd.gt.thresh .or. dabs(accu_d-dble(n)).gt.thresh) then
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! stop
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!endif
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end subroutine get_inv_half_svd
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! ---
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subroutine get_inv_half_nonsymmat_diago(matrix, n, matrix_inv_half, complex_root)
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BEGIN_DOC
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! input: S = matrix
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! output: S^{-1/2} = matrix_inv_half obtained by diagonalization
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!
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! S = VR D VL^T
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! = VR D^{1/2} D^{1/2} VL^T
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! = VR D^{1/2} VL^T VR D^{1/2} VL^T
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! = S^{1/2} S^{1/2} with S = VR D^{1/2} VL^T
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!
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! == > S^{-1/2} = VR D^{-1/2} VL^T
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!
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END_DOC
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implicit none
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integer, intent(in) :: n
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double precision, intent(in) :: matrix(n,n)
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logical, intent(out) :: complex_root
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double precision, intent(out) :: matrix_inv_half(n,n)
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integer :: i, j
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double precision :: accu_d, accu_nd
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double precision, allocatable :: WR(:), WI(:), VL(:,:), VR(:,:), S(:,:), S_diag(:)
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double precision, allocatable :: tmp1(:,:), D_mat(:,:)
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complex_root = .False.
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matrix_inv_half = 0.D0
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print*,'Computing S^{-1/2}'
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allocate(WR(n), WI(n), VL(n,n), VR(n,n))
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call lapack_diag_non_sym(n, matrix, WR, WI, VL, VR)
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allocate(S(n,n))
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call check_biorthog(n, n, VL, VR, accu_d, accu_nd, S)
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print*,'accu_nd S^{-1/2}',accu_nd
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if(accu_nd.gt.1.d-10) then
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complex_root = .True. ! if vectors are not bi-orthogonal return
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print*,'Eigenvectors of S are not bi-orthonormal, skipping S^{-1/2}'
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return
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endif
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allocate(S_diag(n))
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do i = 1, n
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S_diag(i) = 1.d0/dsqrt(S(i,i))
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if(dabs(WI(i)).gt.1.d-20.or.WR(i).lt.0.d0)then ! check that eigenvalues are real and positive
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complex_root = .True.
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print*,'Eigenvalues of S have imaginary part '
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print*,'WR(i),WI(i)',WR(i), WR(i)
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print*,'Skipping S^{-1/2}'
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return
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endif
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enddo
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deallocate(S)
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if(complex_root) return
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! normalization of vectors
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do i = 1, n
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if(S_diag(i).eq.1.d0) cycle
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do j = 1,n
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VL(j,i) *= S_diag(i)
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VR(j,i) *= S_diag(i)
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enddo
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enddo
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deallocate(S_diag)
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allocate(tmp1(n,n), D_mat(n,n))
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D_mat = 0.d0
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do i = 1, n
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D_mat(i,i) = 1.d0/dsqrt(WR(i))
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enddo
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deallocate(WR, WI)
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! tmp1 = VR D^{-1/2}
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tmp1 = 0.d0
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call dgemm( 'N', 'N', n, n, n, 1.d0 &
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, VR, size(VR, 1), D_mat, size(D_mat, 1) &
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, 0.d0, tmp1, size(tmp1, 1) )
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deallocate(VR, D_mat)
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! S^{-1/2} = tmp1 X VL^T
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matrix_inv_half = 0.d0
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call dgemm( 'N', 'T', n, n, n, 1.d0 &
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, tmp1, size(tmp1, 1), VL, size(VL, 1) &
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, 0.d0, matrix_inv_half, size(matrix_inv_half, 1) )
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deallocate(tmp1, VL)
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end
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! ---
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subroutine bi_ortho_s_inv_half(n,leigvec,reigvec,S_nh_inv_half)
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implicit none
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integer, intent(in) :: n
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double precision, intent(in) :: S_nh_inv_half(n,n)
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double precision, intent(inout) :: leigvec(n,n),reigvec(n,n)
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BEGIN_DOC
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! bi-orthonormalization of left and right vectors
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!
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! S = VL^T VR
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!
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! S^{-1/2} S S^{-1/2} = 1 = S^{-1/2} VL^T VR S^{-1/2} = VL_new^T VR_new
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!
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! VL_new = VL (S^{-1/2})^T
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!
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! VR_new = VR S^{^{-1/2}}
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END_DOC
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double precision,allocatable :: vl_tmp(:,:),vr_tmp(:,:)
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print*,'Bi-orthonormalization using S^{-1/2}'
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allocate(vl_tmp(n,n),vr_tmp(n,n))
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vl_tmp = leigvec
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vr_tmp = reigvec
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! VL_new = VL (S^{-1/2})^T
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call dgemm( 'N', 'T', n, n, n, 1.d0 &
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, vl_tmp, size(vl_tmp, 1), S_nh_inv_half, size(S_nh_inv_half, 1) &
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, 0.d0, leigvec, size(leigvec, 1) )
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! VR_new = VR S^{^{-1/2}}
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call dgemm( 'N', 'N', n, n, n, 1.d0 &
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, vr_tmp, size(vr_tmp, 1), S_nh_inv_half, size(S_nh_inv_half, 1) &
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, 0.d0, reigvec, size(reigvec, 1) )
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double precision :: accu_d, accu_nd
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double precision,allocatable :: S(:,:)
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allocate(S(n,n))
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call check_biorthog(n, n, leigvec, reigvec, accu_d, accu_nd, S)
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if(dabs(accu_d - n).gt.1.d-10 .or. accu_nd .gt.1.d-8 )then
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print*,'Pb in bi_ortho_s_inv_half !!'
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print*,'accu_d =',accu_d
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print*,'accu_nd =',accu_nd
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stop
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endif
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end
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