!--------------------------------------------------------------------------------------------------!
!   CP2K: A general program to perform molecular dynamics simulations                              !
!   Copyright (C) 2000 - 2018  CP2K developers group                                               !
!--------------------------------------------------------------------------------------------------!

! **************************************************************************************************
!> \brief Cayley transformation methods
!> \par History
!>       2011.06 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
MODULE ct_methods
   USE cp_dbcsr_cholesky,               ONLY: cp_dbcsr_cholesky_decompose,&
                                              cp_dbcsr_cholesky_invert
   USE cp_dbcsr_diag,                   ONLY: cp_dbcsr_syevd
   USE cp_log_handling,                 ONLY: cp_get_default_logger,&
                                              cp_logger_get_default_unit_nr,&
                                              cp_logger_type
   USE ct_types,                        ONLY: ct_step_env_type
   USE dbcsr_api,                       ONLY: &
        dbcsr_add, dbcsr_add_on_diag, dbcsr_copy, dbcsr_create, dbcsr_desymmetrize, &
        dbcsr_distribution_get, dbcsr_distribution_type, dbcsr_filter, dbcsr_finalize, &
        dbcsr_frobenius_norm, dbcsr_func_inverse, dbcsr_function_of_elements, dbcsr_get_diag, &
        dbcsr_get_info, dbcsr_get_stored_coordinates, dbcsr_hadamard_product, &
        dbcsr_iterator_blocks_left, dbcsr_iterator_next_block, dbcsr_iterator_start, &
        dbcsr_iterator_stop, dbcsr_iterator_type, dbcsr_multiply, dbcsr_nblkcols_total, &
        dbcsr_nblkrows_total, dbcsr_norm, dbcsr_norm_maxabsnorm, dbcsr_release, &
        dbcsr_reserve_block2d, dbcsr_scale, dbcsr_set, dbcsr_set_diag, dbcsr_trace, &
        dbcsr_transposed, dbcsr_type, dbcsr_type_no_symmetry, dbcsr_work_create
   USE input_constants,                 ONLY: &
        cg_dai_yuan, cg_fletcher, cg_fletcher_reeves, cg_hager_zhang, cg_hestenes_stiefel, &
        cg_liu_storey, cg_polak_ribiere, cg_zero, tensor_orthogonal, tensor_up_down
   USE iterate_matrix,                  ONLY: matrix_sqrt_Newton_Schulz
   USE kinds,                           ONLY: dp
   USE machine,                         ONLY: m_walltime
#include "./base/base_uses.f90"

   IMPLICIT NONE

   PRIVATE

   CHARACTER(len=*), PARAMETER, PRIVATE :: moduleN = 'ct_methods'

   ! Public subroutines
   PUBLIC :: ct_step_execute, analytic_line_search, diagonalize_diagonal_blocks

CONTAINS

! **************************************************************************************************
!> \brief Performs Cayley transformation
!> \param cts_env ...
!> \par History
!>       2011.06 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
   SUBROUTINE ct_step_execute(cts_env)

      TYPE(ct_step_env_type)                             :: cts_env

      CHARACTER(len=*), PARAMETER :: routineN = 'ct_step_execute', &
         routineP = moduleN//':'//routineN

      INTEGER                                            :: handle, n, preconditioner_type, unit_nr
      REAL(KIND=dp)                                      :: gap_estimate, safety_margin
      REAL(KIND=dp), ALLOCATABLE, DIMENSION(:)           :: evals
      TYPE(cp_logger_type), POINTER                      :: logger
      TYPE(dbcsr_type)                                   :: matrix_pp, matrix_pq, matrix_qp, &
                                                            matrix_qp_save, matrix_qq, oo1, &
                                                            oo1_sqrt, oo1_sqrt_inv, t_corr, tmp1, &
                                                            u_pp, u_qq

!TYPE(dbcsr_type)                :: rst_x1, rst_x2
!REAL(KIND=dp)                      :: ener_tmp
!TYPE(dbcsr_iterator_type)            :: iter
!INTEGER                            :: iblock_row,iblock_col,&
!                                      iblock_row_size,iblock_col_size
!REAL(KIND=dp), DIMENSION(:,:), POINTER :: data_p

      CALL timeset(routineN, handle)

      logger => cp_get_default_logger()
      IF (logger%para_env%mepos == logger%para_env%source) THEN
         unit_nr = cp_logger_get_default_unit_nr(logger, local=.TRUE.)
      ELSE
         unit_nr = -1
      ENDIF

      ! check if all input is in place and flags are consistent
      IF (cts_env%update_q .AND. (.NOT. cts_env%update_p)) THEN
         CPABORT("q-update is possible only with p-update")
      ENDIF

      IF (cts_env%tensor_type .EQ. tensor_up_down) THEN
         CPABORT("riccati is not implemented for biorthogonal basis")
      ENDIF

      IF (.NOT. ASSOCIATED(cts_env%matrix_ks)) THEN
         CPABORT("KS matrix is not associated")
      ENDIF

      IF (cts_env%use_virt_orbs .AND. (.NOT. cts_env%use_occ_orbs)) THEN
         CPABORT("virtual orbs can be used only with occupied orbs")
      ENDIF

      IF (cts_env%use_occ_orbs) THEN
         IF (.NOT. ASSOCIATED(cts_env%matrix_t)) THEN
            CPABORT("T matrix is not associated")
         ENDIF
         IF (.NOT. ASSOCIATED(cts_env%matrix_qp_template)) THEN
            CPABORT("QP template is not associated")
         ENDIF
         IF (.NOT. ASSOCIATED(cts_env%matrix_pq_template)) THEN
            CPABORT("PQ template is not associated")
         ENDIF
      ENDIF

      IF (cts_env%use_virt_orbs) THEN
         IF (.NOT. ASSOCIATED(cts_env%matrix_v)) THEN
            CPABORT("V matrix is not associated")
         ENDIF
      ELSE
         IF (.NOT. ASSOCIATED(cts_env%matrix_p)) THEN
            CPABORT("P matrix is not associated")
         ENDIF
      ENDIF

      IF (cts_env%tensor_type .NE. tensor_up_down .AND. &
          cts_env%tensor_type .NE. tensor_orthogonal) THEN
         CPABORT("illegal tensor flag")
      ENDIF

      ! start real calculations
      IF (cts_env%use_occ_orbs) THEN

         ! create matrices for various ks blocks
         CALL dbcsr_create(matrix_pp, &
                           template=cts_env%p_index_up, &
                           matrix_type=dbcsr_type_no_symmetry)
         CALL dbcsr_create(matrix_qp, &
                           template=cts_env%matrix_qp_template, &
                           matrix_type=dbcsr_type_no_symmetry)
         CALL dbcsr_create(matrix_qq, &
                           template=cts_env%q_index_up, &
                           matrix_type=dbcsr_type_no_symmetry)
         CALL dbcsr_create(matrix_pq, &
                           template=cts_env%matrix_pq_template, &
                           matrix_type=dbcsr_type_no_symmetry)

         ! create the residue matrix
         CALL dbcsr_create(cts_env%matrix_res, &
                           template=cts_env%matrix_qp_template)

         CALL assemble_ks_qp_blocks(cts_env%matrix_ks, &
                                    cts_env%matrix_p, &
                                    cts_env%matrix_t, &
                                    cts_env%matrix_v, &
                                    cts_env%q_index_down, &
                                    cts_env%p_index_up, &
                                    cts_env%q_index_up, &
                                    matrix_pp, &
                                    matrix_qq, &
                                    matrix_qp, &
                                    matrix_pq, &
                                    cts_env%tensor_type, &
                                    cts_env%use_virt_orbs, &
                                    cts_env%eps_filter)

         ! create a matrix of single-excitation amplitudes
         CALL dbcsr_create(cts_env%matrix_x, &
                           template=cts_env%matrix_qp_template)
         IF (ASSOCIATED(cts_env%matrix_x_guess)) THEN
            CALL dbcsr_copy(cts_env%matrix_x, &
                            cts_env%matrix_x_guess)
            IF (cts_env%tensor_type .EQ. tensor_orthogonal) THEN
               ! bring x from contravariant-covariant representation
               ! to the orthogonal/cholesky representation
               ! use res as temporary storage
               CALL dbcsr_multiply("N", "N", 1.0_dp, cts_env%q_index_down, &
                                   cts_env%matrix_x, 0.0_dp, cts_env%matrix_res, &
                                   filter_eps=cts_env%eps_filter)
               CALL dbcsr_multiply("N", "N", 1.0_dp, cts_env%matrix_res, &
                                   cts_env%p_index_up, 0.0_dp, &
                                   cts_env%matrix_x, &
                                   filter_eps=cts_env%eps_filter)
            ENDIF
         ELSE
            ! set amplitudes to zero
            CALL dbcsr_set(cts_env%matrix_x, 0.0_dp)
         ENDIF

         !SELECT CASE (cts_env%preconditioner_type)
         !CASE (prec_eigenvector_blocks,prec_eigenvector_full)
         preconditioner_type = 1
         safety_margin = 2.0_dp
         gap_estimate = 0.0001_dp
         SELECT CASE (preconditioner_type)
         CASE (1, 2)
!RZK-warning diagonalization works only with orthogonal tensor!!!
            ! find a better basis by diagonalizing diagonal blocks
            ! first pp
            CALL dbcsr_create(u_pp, template=matrix_pp, &
                              matrix_type=dbcsr_type_no_symmetry)
            !IF (cts_env%preconditioner_type.eq.prec_eigenvector_full) THEN
            IF (.TRUE.) THEN
               CALL dbcsr_get_info(matrix_pp, nfullrows_total=n)
               ALLOCATE (evals(n))
               CALL cp_dbcsr_syevd(matrix_pp, u_pp, evals, &
                                   cts_env%para_env, cts_env%blacs_env)
               DEALLOCATE (evals)
            ELSE
               CALL diagonalize_diagonal_blocks(matrix_pp, u_pp)
            ENDIF
            ! and now qq
            CALL dbcsr_create(u_qq, template=matrix_qq, &
                              matrix_type=dbcsr_type_no_symmetry)
            !IF (cts_env%preconditioner_type.eq.prec_eigenvector_full) THEN
            IF (.TRUE.) THEN
               CALL dbcsr_get_info(matrix_qq, nfullrows_total=n)
               ALLOCATE (evals(n))
               CALL cp_dbcsr_syevd(matrix_qq, u_qq, evals, &
                                   cts_env%para_env, cts_env%blacs_env)
               DEALLOCATE (evals)
            ELSE
               CALL diagonalize_diagonal_blocks(matrix_qq, u_qq)
            ENDIF

            ! apply the transformation to all matrices
            CALL matrix_forward_transform(matrix_pp, u_pp, u_pp, &
                                          cts_env%eps_filter)
            CALL matrix_forward_transform(matrix_qq, u_qq, u_qq, &
                                          cts_env%eps_filter)
            CALL matrix_forward_transform(matrix_qp, u_qq, u_pp, &
                                          cts_env%eps_filter)
            CALL matrix_forward_transform(matrix_pq, u_pp, u_qq, &
                                          cts_env%eps_filter)
            CALL matrix_forward_transform(cts_env%matrix_x, u_qq, u_pp, &
                                          cts_env%eps_filter)

            IF (cts_env%max_iter .GE. 0) THEN

               CALL solve_riccati_equation( &
                  pp=matrix_pp, &
                  qq=matrix_qq, &
                  qp=matrix_qp, &
                  pq=matrix_pq, &
                  x=cts_env%matrix_x, &
                  res=cts_env%matrix_res, &
                  neglect_quadratic_term=cts_env%neglect_quadratic_term, &
                  conjugator=cts_env%conjugator, &
                  max_iter=cts_env%max_iter, &
                  eps_convergence=cts_env%eps_convergence, &
                  eps_filter=cts_env%eps_filter, &
                  converged=cts_env%converged)

               IF (cts_env%converged) THEN
                  !IF (unit_nr>0) THEN
                  !   WRITE(unit_nr,*)
                  !   WRITE(unit_nr,'(T6,A)') &
                  !         "RICCATI equations solved"
                  !   CALL m_flush(unit_nr)
                  !ENDIF
               ELSE
                  CPABORT("RICCATI: CG algorithm has NOT converged")
               ENDIF

            ENDIF

            IF (cts_env%calculate_energy_corr) THEN

               CALL dbcsr_trace(matrix_qp, cts_env%matrix_x, cts_env%energy_correction)

            ENDIF

            CALL dbcsr_release(matrix_pp)
            CALL dbcsr_release(matrix_qp)
            CALL dbcsr_release(matrix_qq)
            CALL dbcsr_release(matrix_pq)

            ! back-transform to the original basis
            CALL matrix_backward_transform(cts_env%matrix_x, u_qq, &
                                           u_pp, cts_env%eps_filter)

            CALL dbcsr_release(u_qq)
            CALL dbcsr_release(u_pp)

            !CASE (prec_cholesky_inverse)
         CASE (3)

! RZK-warning implemented only for orthogonal tensors!!!
! generalization to up_down should be easy
            CALL dbcsr_create(u_pp, template=matrix_pp, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_copy(u_pp, matrix_pp)
            CALL dbcsr_scale(u_pp, -1.0_dp)
            CALL dbcsr_add_on_diag(u_pp, &
                                   ABS(safety_margin*gap_estimate))
            CALL cp_dbcsr_cholesky_decompose(u_pp, &
                                             para_env=cts_env%para_env, &
                                             blacs_env=cts_env%blacs_env)
            CALL cp_dbcsr_cholesky_invert(u_pp, &
                                          para_env=cts_env%para_env, &
                                          blacs_env=cts_env%blacs_env, &
                                          upper_to_full=.TRUE.)
            !CALL dbcsr_scale(u_pp,-1.0_dp)

            CALL dbcsr_create(u_qq, template=matrix_qq, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_copy(u_qq, matrix_qq)
            CALL dbcsr_add_on_diag(u_qq, &
                                   ABS(safety_margin*gap_estimate))
            CALL cp_dbcsr_cholesky_decompose(u_qq, &
                                             para_env=cts_env%para_env, &
                                             blacs_env=cts_env%blacs_env)
            CALL cp_dbcsr_cholesky_invert(u_qq, &
                                          para_env=cts_env%para_env, &
                                          blacs_env=cts_env%blacs_env, &
                                          upper_to_full=.TRUE.)

            ! transform all riccati matrices (left-right preconditioner)
            CALL dbcsr_create(tmp1, template=matrix_qq, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_multiply("N", "N", 1.0_dp, u_qq, &
                                matrix_qq, 0.0_dp, tmp1, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_copy(matrix_qq, tmp1)
            CALL dbcsr_release(tmp1)

            CALL dbcsr_create(tmp1, template=matrix_pp, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_multiply("N", "N", 1.0_dp, matrix_pp, &
                                u_pp, 0.0_dp, tmp1, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_copy(matrix_pp, tmp1)
            CALL dbcsr_release(tmp1)

            CALL dbcsr_create(matrix_qp_save, template=matrix_qp, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_copy(matrix_qp_save, matrix_qp)

            CALL dbcsr_create(tmp1, template=matrix_qp, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_multiply("N", "N", 1.0_dp, matrix_qp, &
                                u_pp, 0.0_dp, tmp1, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_multiply("N", "N", 1.0_dp, u_qq, tmp1, &
                                0.0_dp, matrix_qp, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_release(tmp1)
!CALL dbcsr_print(matrix_qq)
!CALL dbcsr_print(matrix_qp)
!CALL dbcsr_print(matrix_pp)

            IF (cts_env%max_iter .GE. 0) THEN

               CALL solve_riccati_equation( &
                  pp=matrix_pp, &
                  qq=matrix_qq, &
                  qp=matrix_qp, &
                  pq=matrix_pq, &
                  oo=u_pp, &
                  vv=u_qq, &
                  x=cts_env%matrix_x, &
                  res=cts_env%matrix_res, &
                  neglect_quadratic_term=cts_env%neglect_quadratic_term, &
                  conjugator=cts_env%conjugator, &
                  max_iter=cts_env%max_iter, &
                  eps_convergence=cts_env%eps_convergence, &
                  eps_filter=cts_env%eps_filter, &
                  converged=cts_env%converged)

               IF (cts_env%converged) THEN
                  !IF (unit_nr>0) THEN
                  !   WRITE(unit_nr,*)
                  !   WRITE(unit_nr,'(T6,A)') &
                  !         "RICCATI equations solved"
                  !   CALL m_flush(unit_nr)
                  !ENDIF
               ELSE
                  CPABORT("RICCATI: CG algorithm has NOT converged")
               ENDIF

            ENDIF

            IF (cts_env%calculate_energy_corr) THEN

               CALL dbcsr_trace(matrix_qp_save, cts_env%matrix_x, cts_env%energy_correction)

            ENDIF
            CALL dbcsr_release(matrix_qp_save)

            CALL dbcsr_release(matrix_pp)
            CALL dbcsr_release(matrix_qp)
            CALL dbcsr_release(matrix_qq)
            CALL dbcsr_release(matrix_pq)

            CALL dbcsr_release(u_qq)
            CALL dbcsr_release(u_pp)

         CASE DEFAULT
            CPABORT("illegal preconditioner type")
         END SELECT ! preconditioner type

         IF (cts_env%update_p) THEN

            IF (cts_env%tensor_type .EQ. tensor_up_down) THEN
               CPABORT("orbital update is NYI for this tensor type")
            ENDIF

            ! transform occupied orbitals
            ! in a way that preserves the overlap metric
            CALL dbcsr_create(oo1, &
                              template=cts_env%p_index_up, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_create(oo1_sqrt_inv, &
                              template=oo1)
            CALL dbcsr_create(oo1_sqrt, &
                              template=oo1)

            ! Compute (1+tr(X).X)^(-1/2)_up_down
            CALL dbcsr_multiply("T", "N", 1.0_dp, cts_env%matrix_x, &
                                cts_env%matrix_x, 0.0_dp, oo1, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_add_on_diag(oo1, 1.0_dp)
            CALL matrix_sqrt_Newton_Schulz(oo1_sqrt, &
                                           oo1_sqrt_inv, &
                                           oo1, &
                                           !if cholesky is used then sqrt
                                           !guess cannot be provided
                                           !matrix_sqrt_inv_guess=cts_env%p_index_up,&
                                           !matrix_sqrt_guess=cts_env%p_index_down,&
                                           threshold=cts_env%eps_filter, &
                                           order=cts_env%order_lanczos, &
                                           eps_lanczos=cts_env%eps_lancsoz, &
                                           max_iter_lanczos=cts_env%max_iter_lanczos)
            CALL dbcsr_multiply("N", "N", 1.0_dp, cts_env%p_index_up, &
                                oo1_sqrt_inv, 0.0_dp, oo1, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_multiply("N", "N", 1.0_dp, oo1, &
                                cts_env%p_index_down, 0.0_dp, oo1_sqrt, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_release(oo1)
            CALL dbcsr_release(oo1_sqrt_inv)

            ! bring x to contravariant-covariant representation now
            CALL dbcsr_create(matrix_qp, &
                              template=cts_env%matrix_qp_template, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_multiply("N", "N", 1.0_dp, cts_env%q_index_up, &
                                cts_env%matrix_x, 0.0_dp, matrix_qp, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_multiply("N", "N", 1.0_dp, matrix_qp, &
                                cts_env%p_index_down, 0.0_dp, &
                                cts_env%matrix_x, &
                                filter_eps=cts_env%eps_filter)
            CALL dbcsr_release(matrix_qp)

            ! update T=T+X or T=T+V.X (whichever is appropriate)
            CALL dbcsr_create(t_corr, template=cts_env%matrix_t)
            IF (cts_env%use_virt_orbs) THEN
               CALL dbcsr_multiply("N", "N", 1.0_dp, cts_env%matrix_v, &
                                   cts_env%matrix_x, 0.0_dp, t_corr, &
                                   filter_eps=cts_env%eps_filter)
               CALL dbcsr_add(cts_env%matrix_t, t_corr, &
                              1.0_dp, 1.0_dp)
            ELSE
               CALL dbcsr_add(cts_env%matrix_t, cts_env%matrix_x, &
                              1.0_dp, 1.0_dp)
            ENDIF
            ! adjust T so the metric is preserved: T=(T+X).(1+tr(X).X)^(-1/2)
            CALL dbcsr_multiply("N", "N", 1.0_dp, cts_env%matrix_t, oo1_sqrt, &
                                0.0_dp, t_corr, filter_eps=cts_env%eps_filter)
            CALL dbcsr_copy(cts_env%matrix_t, t_corr)

            CALL dbcsr_release(t_corr)
            CALL dbcsr_release(oo1_sqrt)

         ELSE ! do not update p

            IF (cts_env%tensor_type .EQ. tensor_orthogonal) THEN
               ! bring x to contravariant-covariant representation
               CALL dbcsr_create(matrix_qp, &
                                 template=cts_env%matrix_qp_template, &
                                 matrix_type=dbcsr_type_no_symmetry)
               CALL dbcsr_multiply("N", "N", 1.0_dp, cts_env%q_index_up, &
                                   cts_env%matrix_x, 0.0_dp, matrix_qp, &
                                   filter_eps=cts_env%eps_filter)
               CALL dbcsr_multiply("N", "N", 1.0_dp, matrix_qp, &
                                   cts_env%p_index_down, 0.0_dp, &
                                   cts_env%matrix_x, &
                                   filter_eps=cts_env%eps_filter)
               CALL dbcsr_release(matrix_qp)
            ENDIF

         ENDIF

      ELSE
         CPABORT("illegal occ option")
      ENDIF

      CALL timestop(handle)

   END SUBROUTINE ct_step_execute

! **************************************************************************************************
!> \brief computes oo, ov, vo, and vv blocks of the ks matrix
!> \param ks ...
!> \param p ...
!> \param t ...
!> \param v ...
!> \param q_index_down ...
!> \param p_index_up ...
!> \param q_index_up ...
!> \param pp ...
!> \param qq ...
!> \param qp ...
!> \param pq ...
!> \param tensor_type ...
!> \param use_virt_orbs ...
!> \param eps_filter ...
!> \par History
!>       2011.06 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
   SUBROUTINE assemble_ks_qp_blocks(ks, p, t, v, q_index_down, &
                                    p_index_up, q_index_up, pp, qq, qp, pq, tensor_type, use_virt_orbs, eps_filter)

      TYPE(dbcsr_type), INTENT(IN)                       :: ks, p, t, v, q_index_down, p_index_up, &
                                                            q_index_up
      TYPE(dbcsr_type), INTENT(OUT)                      :: pp, qq, qp, pq
      INTEGER, INTENT(IN)                                :: tensor_type
      LOGICAL, INTENT(IN)                                :: use_virt_orbs
      REAL(KIND=dp), INTENT(IN)                          :: eps_filter

      CHARACTER(len=*), PARAMETER :: routineN = 'assemble_ks_qp_blocks', &
         routineP = moduleN//':'//routineN

      INTEGER                                            :: handle
      LOGICAL                                            :: library_fixed
      TYPE(dbcsr_type)                                   :: kst, ksv, no, on, oo, q_index_up_nosym, &
                                                            sp, spf, t_or, v_or

      CALL timeset(routineN, handle)

      IF (use_virt_orbs) THEN

         ! orthogonalize the orbitals
         CALL dbcsr_create(t_or, template=t)
         CALL dbcsr_create(v_or, template=v)
         CALL dbcsr_multiply("N", "N", 1.0_dp, t, p_index_up, &
                             0.0_dp, t_or, filter_eps=eps_filter)
         CALL dbcsr_multiply("N", "N", 1.0_dp, v, q_index_up, &
                             0.0_dp, v_or, filter_eps=eps_filter)

         ! KS.T
         CALL dbcsr_create(kst, template=t)
         CALL dbcsr_multiply("N", "N", 1.0_dp, ks, t_or, &
                             0.0_dp, kst, filter_eps=eps_filter)
         ! pp=tr(T)*KS.T
         CALL dbcsr_multiply("T", "N", 1.0_dp, t_or, kst, &
                             0.0_dp, pp, filter_eps=eps_filter)
         ! qp=tr(V)*KS.T
         CALL dbcsr_multiply("T", "N", 1.0_dp, v_or, kst, &
                             0.0_dp, qp, filter_eps=eps_filter)
         CALL dbcsr_release(kst)

         ! KS.V
         CALL dbcsr_create(ksv, template=v)
         CALL dbcsr_multiply("N", "N", 1.0_dp, ks, v_or, &
                             0.0_dp, ksv, filter_eps=eps_filter)
         ! tr(T)*KS.V
         CALL dbcsr_multiply("T", "N", 1.0_dp, t_or, ksv, &
                             0.0_dp, pq, filter_eps=eps_filter)
         ! tr(V)*KS.V
         CALL dbcsr_multiply("T", "N", 1.0_dp, v_or, ksv, &
                             0.0_dp, qq, filter_eps=eps_filter)
         CALL dbcsr_release(ksv)

         CALL dbcsr_release(t_or)
         CALL dbcsr_release(v_or)

      ELSE ! no virtuals, use projected AOs

! THIS PROCEDURE HAS NOT BEEN UPDATED FOR CHOLESKY p/q_index_up/down
         CALL dbcsr_create(sp, template=q_index_down, &
                           matrix_type=dbcsr_type_no_symmetry)
         CALL dbcsr_create(spf, template=q_index_down, &
                           matrix_type=dbcsr_type_no_symmetry)

         ! qp=KS*T
         CALL dbcsr_multiply("N", "N", 1.0_dp, ks, t, 0.0_dp, qp, &
                             filter_eps=eps_filter)
         ! pp=tr(T)*KS.T
         CALL dbcsr_multiply("T", "N", 1.0_dp, t, qp, 0.0_dp, pp, &
                             filter_eps=eps_filter)
         ! sp=-S_*P
         CALL dbcsr_multiply("N", "N", -1.0_dp, q_index_down, p, 0.0_dp, sp, &
                             filter_eps=eps_filter)

         ! sp=1/S^-S_.P
         SELECT CASE (tensor_type)
         CASE (tensor_up_down)
            CALL dbcsr_add_on_diag(sp, 1.0_dp)
         CASE (tensor_orthogonal)
            CALL dbcsr_create(q_index_up_nosym, template=q_index_up, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_desymmetrize(q_index_up, q_index_up_nosym)
            CALL dbcsr_add(sp, q_index_up_nosym, 1.0_dp, 1.0_dp)
            CALL dbcsr_release(q_index_up_nosym)
         END SELECT

         ! spf=(1/S^-S_.P)*KS
         CALL dbcsr_multiply("N", "N", 1.0_dp, sp, ks, 0.0_dp, spf, &
                             filter_eps=eps_filter)

         ! qp=spf*T
         CALL dbcsr_multiply("N", "N", 1.0_dp, spf, t, 0.0_dp, qp, &
                             filter_eps=eps_filter)

         SELECT CASE (tensor_type)
         CASE (tensor_up_down)
            ! pq=tr(qp)
            CALL dbcsr_transposed(pq, qp, transpose_distribution=.FALSE.)
         CASE (tensor_orthogonal)
            ! pq=sig^.tr(qp)
            CALL dbcsr_multiply("N", "T", 1.0_dp, p_index_up, qp, 0.0_dp, pq, &
                                filter_eps=eps_filter)
            library_fixed = .FALSE.
            IF (library_fixed) THEN
               CALL dbcsr_transposed(qp, pq, transpose_distribution=.FALSE.)
            ELSE
               CALL dbcsr_create(no, template=qp, &
                                 matrix_type=dbcsr_type_no_symmetry)
               CALL dbcsr_multiply("N", "N", 1.0_dp, qp, p_index_up, 0.0_dp, no, &
                                   filter_eps=eps_filter)
               CALL dbcsr_copy(qp, no)
               CALL dbcsr_release(no)
            ENDIF
         END SELECT

         ! qq=spf*tr(sp)
         CALL dbcsr_multiply("N", "T", 1.0_dp, spf, sp, 0.0_dp, qq, &
                             filter_eps=eps_filter)

         SELECT CASE (tensor_type)
         CASE (tensor_up_down)

            CALL dbcsr_create(oo, template=pp, &
                              matrix_type=dbcsr_type_no_symmetry)
            CALL dbcsr_create(no, template=qp, &
                              matrix_type=dbcsr_type_no_symmetry)

            ! first index up
            CALL dbcsr_multiply("N", "N", 1.0_dp, q_index_up, qq, 0.0_dp, spf, &
                                filter_eps=eps_filter)
            CALL dbcsr_copy(qq, spf)
            CALL dbcsr_multiply("N", "N", 1.0_dp, q_index_up, qp, 0.0_dp, no, &
                                filter_eps=eps_filter)
            CALL dbcsr_copy(qp, no)
            CALL dbcsr_multiply("N", "N", 1.0_dp, p_index_up, pp, 0.0_dp, oo, &
                                filter_eps=eps_filter)
            CALL dbcsr_copy(pp, oo)
            CALL dbcsr_multiply("N", "N", 1.0_dp, p_index_up, pq, 0.0_dp, on, &
                                filter_eps=eps_filter)
            CALL dbcsr_copy(pq, on)

            CALL dbcsr_release(no)
            CALL dbcsr_release(oo)

         CASE (tensor_orthogonal)

            CALL dbcsr_create(oo, template=pp, &
                              matrix_type=dbcsr_type_no_symmetry)

            ! both indeces up in the pp block
            CALL dbcsr_multiply("N", "N", 1.0_dp, p_index_up, pp, 0.0_dp, oo, &
                                filter_eps=eps_filter)
            CALL dbcsr_multiply("N", "N", 1.0_dp, oo, p_index_up, 0.0_dp, pp, &
                                filter_eps=eps_filter)

            CALL dbcsr_release(oo)

         END SELECT

         CALL dbcsr_release(sp)
         CALL dbcsr_release(spf)

      ENDIF

      CALL timestop(handle)

   END SUBROUTINE assemble_ks_qp_blocks

! **************************************************************************************************
!> \brief Solves the generalized Riccati or Sylvester eqation
!>        using the preconditioned conjugate gradient algorithm
!>          qp + qq.x.oo - vv.x.pp - vv.x.pq.x.oo = 0 [oo and vv are optional]
!>          qp + qq.x - x.pp - x.pq.x = 0
!> \param pp ...
!> \param qq ...
!> \param qp ...
!> \param pq ...
!> \param oo ...
!> \param vv ...
!> \param x ...
!> \param res ...
!> \param neglect_quadratic_term ...
!> \param conjugator ...
!> \param max_iter ...
!> \param eps_convergence ...
!> \param eps_filter ...
!> \param converged ...
!> \par History
!>       2011.06 created [Rustam Z Khaliullin]
!>       2011.11 generalized [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
   RECURSIVE SUBROUTINE solve_riccati_equation(pp, qq, qp, pq, oo, vv, x, res, &
                                               neglect_quadratic_term, &
                                               conjugator, max_iter, eps_convergence, eps_filter, &
                                               converged)

      TYPE(dbcsr_type), INTENT(IN)                       :: pp, qq
      TYPE(dbcsr_type), INTENT(INOUT)                    :: qp
      TYPE(dbcsr_type), INTENT(IN)                       :: pq
      TYPE(dbcsr_type), INTENT(IN), OPTIONAL             :: oo, vv
      TYPE(dbcsr_type), INTENT(INOUT)                    :: x
      TYPE(dbcsr_type), INTENT(OUT)                      :: res
      LOGICAL, INTENT(IN)                                :: neglect_quadratic_term
      INTEGER, INTENT(IN)                                :: conjugator, max_iter
      REAL(KIND=dp), INTENT(IN)                          :: eps_convergence, eps_filter
      LOGICAL, INTENT(OUT)                               :: converged

      CHARACTER(len=*), PARAMETER :: routineN = 'solve_riccati_equation', &
         routineP = moduleN//':'//routineN

      INTEGER                                            :: handle, istep, iteration, nsteps, &
                                                            unit_nr, update_prec_freq
      LOGICAL                                            :: prepare_to_exit, present_oo, present_vv, &
                                                            quadratic_term, restart_conjugator
      REAL(KIND=dp)                                      :: best_norm, best_step_size, beta, c0, c1, &
                                                            c2, c3, denom, kappa, numer, &
                                                            obj_function, t1, t2, tau
      REAL(KIND=dp), DIMENSION(3)                        :: step_size
      TYPE(cp_logger_type), POINTER                      :: logger
      TYPE(dbcsr_type)                                   :: aux1, aux2, grad, m, n, oo1, oo2, prec, &
                                                            res_trial, step, step_oo, vv_step

!TYPE(dbcsr_type)                      :: qqqq, pppp, zero_pq, zero_qp

      CALL timeset(routineN, handle)

      logger => cp_get_default_logger()
      IF (logger%para_env%mepos == logger%para_env%source) THEN
         unit_nr = cp_logger_get_default_unit_nr(logger, local=.TRUE.)
      ELSE
         unit_nr = -1
      ENDIF

      t1 = m_walltime()

!IF (level.gt.5) THEN
!  CPErrorMessage(cp_failure_level,routineP,"recursion level is too high")
!  CPPrecondition(.FALSE.,cp_failure_level,routineP,failure)
!ENDIF
!IF (unit_nr>0) THEN
!   WRITE(unit_nr,*) &
!      "========== LEVEL ",level,"=========="
!ENDIF
!CALL dbcsr_print(qq)
!CALL dbcsr_print(pp)
!CALL dbcsr_print(qp)
!!CALL dbcsr_print(pq)
!IF (unit_nr>0) THEN
!   WRITE(unit_nr,*) &
!      "====== END LEVEL ",level,"=========="
!ENDIF

      quadratic_term = .NOT. neglect_quadratic_term
      present_oo = PRESENT(oo)
      present_vv = PRESENT(vv)

      ! create aux1 matrix and init
      CALL dbcsr_create(aux1, template=pp)
      CALL dbcsr_copy(aux1, pp)
      CALL dbcsr_scale(aux1, -1.0_dp)

      ! create aux2 matrix and init
      CALL dbcsr_create(aux2, template=qq)
      CALL dbcsr_copy(aux2, qq)

      ! create the gradient matrix and init
      CALL dbcsr_create(grad, template=x)
      CALL dbcsr_set(grad, 0.0_dp)

      ! create a preconditioner
      ! RZK-warning how to apply it to up_down tensor?
      CALL dbcsr_create(prec, template=x)
      !CALL create_preconditioner(prec,aux1,aux2,qp,res,tensor_type,eps_filter)
      !CALL dbcsr_set(prec,1.0_dp)

      ! create the step matrix and init
      CALL dbcsr_create(step, template=x)
      !CALL dbcsr_hadamard_product(prec,grad,step)
      !CALL dbcsr_scale(step,-1.0_dp)

      CALL dbcsr_create(n, template=x)
      CALL dbcsr_create(m, template=x)
      CALL dbcsr_create(oo1, template=pp)
      CALL dbcsr_create(oo2, template=pp)
      CALL dbcsr_create(res_trial, template=res)
      CALL dbcsr_create(vv_step, template=res)
      CALL dbcsr_create(step_oo, template=res)

      ! start conjugate gradient iterations
      iteration = 0
      converged = .FALSE.
      prepare_to_exit = .FALSE.
      beta = 0.0_dp
      best_step_size = 0.0_dp
      best_norm = 1.0E+100_dp
      !ecorr=0.0_dp
      !change_ecorr=0.0_dp
      restart_conjugator = .FALSE.
      update_prec_freq = 20
      DO

         ! (re)-compute the residuals
         IF (iteration .EQ. 0) THEN
            CALL dbcsr_copy(res, qp)
            IF (present_oo) THEN
               CALL dbcsr_multiply("N", "N", +1.0_dp, qq, x, 0.0_dp, res_trial, &
                                   filter_eps=eps_filter)
               CALL dbcsr_multiply("N", "N", +1.0_dp, res_trial, oo, 1.0_dp, res, &
                                   filter_eps=eps_filter)
            ELSE
               CALL dbcsr_multiply("N", "N", +1.0_dp, qq, x, 1.0_dp, res, &
                                   filter_eps=eps_filter)
            ENDIF
            IF (present_vv) THEN
               CALL dbcsr_multiply("N", "N", -1.0_dp, x, pp, 0.0_dp, res_trial, &
                                   filter_eps=eps_filter)
               CALL dbcsr_multiply("N", "N", +1.0_dp, vv, res_trial, 1.0_dp, res, &
                                   filter_eps=eps_filter)
            ELSE
               CALL dbcsr_multiply("N", "N", -1.0_dp, x, pp, 1.0_dp, res, &
                                   filter_eps=eps_filter)
            ENDIF
            IF (quadratic_term) THEN
               IF (present_oo) THEN
                  CALL dbcsr_multiply("N", "N", +1.0_dp, pq, x, 0.0_dp, oo1, &
                                      filter_eps=eps_filter)
                  CALL dbcsr_multiply("N", "N", +1.0_dp, oo1, oo, 0.0_dp, oo2, &
                                      filter_eps=eps_filter)
               ELSE
                  CALL dbcsr_multiply("N", "N", +1.0_dp, pq, x, 0.0_dp, oo2, &
                                      filter_eps=eps_filter)
               ENDIF
               IF (present_vv) THEN
                  CALL dbcsr_multiply("N", "N", -1.0_dp, x, oo2, 0.0_dp, res_trial, &
                                      filter_eps=eps_filter)
                  CALL dbcsr_multiply("N", "N", +1.0_dp, vv, res_trial, 1.0_dp, res, &
                                      filter_eps=eps_filter)
               ELSE
                  CALL dbcsr_multiply("N", "N", -1.0_dp, x, oo2, 1.0_dp, res, &
                                      filter_eps=eps_filter)
               ENDIF
            ENDIF
            CALL dbcsr_norm(res, dbcsr_norm_maxabsnorm, norm_scalar=best_norm)
         ELSE
            CALL dbcsr_add(res, m, 1.0_dp, best_step_size)
            CALL dbcsr_add(res, n, 1.0_dp, -best_step_size*best_step_size)
            CALL dbcsr_filter(res, eps_filter)
         ENDIF

         ! check convergence and other exit criteria
         converged = (best_norm .LT. eps_convergence)
         IF (converged .OR. (iteration .GE. max_iter)) THEN
            prepare_to_exit = .TRUE.
         ENDIF

         IF (.NOT. prepare_to_exit) THEN

            ! update aux1=-pp-pq.x.oo and aux2=qq-vv.x.pq
            IF (quadratic_term) THEN
               IF (iteration .EQ. 0) THEN
                  IF (present_oo) THEN
                     CALL dbcsr_multiply("N", "N", -1.0_dp, pq, x, 0.0_dp, oo1, &
                                         filter_eps=eps_filter)
                     CALL dbcsr_multiply("N", "N", +1.0_dp, oo1, oo, 1.0_dp, aux1, &
                                         filter_eps=eps_filter)
                  ELSE
                     CALL dbcsr_multiply("N", "N", -1.0_dp, pq, x, 1.0_dp, aux1, &
                                         filter_eps=eps_filter)
                  ENDIF
                  IF (present_vv) THEN
                     CALL dbcsr_multiply("N", "N", -1.0_dp, vv, x, 0.0_dp, res_trial, &
                                         filter_eps=eps_filter)
                     CALL dbcsr_multiply("N", "N", +1.0_dp, res_trial, pq, 1.0_dp, aux2, &
                                         filter_eps=eps_filter)
                  ELSE
                     CALL dbcsr_multiply("N", "N", -1.0_dp, x, pq, 1.0_dp, aux2, &
                                         filter_eps=eps_filter)
                  ENDIF
               ELSE
                  IF (present_oo) THEN
                     CALL dbcsr_multiply("N", "N", -best_step_size, pq, step_oo, 1.0_dp, aux1, &
                                         filter_eps=eps_filter)
                  ELSE
                     CALL dbcsr_multiply("N", "N", -best_step_size, pq, step, 1.0_dp, aux1, &
                                         filter_eps=eps_filter)
                  ENDIF
                  IF (present_vv) THEN
                     CALL dbcsr_multiply("N", "N", -best_step_size, vv_step, pq, 1.0_dp, aux2, &
                                         filter_eps=eps_filter)
                  ELSE
                     CALL dbcsr_multiply("N", "N", -best_step_size, step, pq, 1.0_dp, aux2, &
                                         filter_eps=eps_filter)
                  ENDIF
               ENDIF
            ENDIF

            ! recompute the gradient, do not update it yet
            ! use m matrix as a temporary storage
            ! grad=t(vv).res.t(aux1)+t(aux2).res.t(oo)
            IF (present_vv) THEN
               CALL dbcsr_multiply("N", "T", 1.0_dp, res, aux1, 0.0_dp, res_trial, &
                                   filter_eps=eps_filter)
               CALL dbcsr_multiply("T", "N", 1.0_dp, vv, res_trial, 0.0_dp, m, &
                                   filter_eps=eps_filter)
            ELSE
               CALL dbcsr_multiply("N", "T", 1.0_dp, res, aux1, 0.0_dp, m, &
                                   filter_eps=eps_filter)
            ENDIF
            IF (present_oo) THEN
               CALL dbcsr_multiply("T", "N", 1.0_dp, aux1, res, 0.0_dp, res_trial, &
                                   filter_eps=eps_filter)
               CALL dbcsr_multiply("N", "T", 1.0_dp, res_trial, oo, 1.0_dp, m, &
                                   filter_eps=eps_filter)
            ELSE
               CALL dbcsr_multiply("T", "N", 1.0_dp, aux2, res, 1.0_dp, m, &
                                   filter_eps=eps_filter)
            ENDIF

            ! compute preconditioner
            !IF (iteration.eq.0.OR.(mod(iteration,update_prec_freq).eq.0)) THEN
            IF (iteration .EQ. 0) THEN
               CALL create_preconditioner(prec, aux1, aux2, eps_filter)
               !restart_conjugator=.TRUE.
!CALL dbcsr_set(prec,1.0_dp)
!CALL dbcsr_print(prec)
            ENDIF

            ! compute the conjugation coefficient - beta
            IF ((iteration .EQ. 0) .OR. restart_conjugator) THEN
               beta = 0.0_dp
            ELSE
               restart_conjugator = .FALSE.
               SELECT CASE (conjugator)
               CASE (cg_hestenes_stiefel)
                  CALL dbcsr_add(grad, m, -1.0_dp, 1.0_dp)
                  CALL dbcsr_hadamard_product(prec, grad, n)
                  CALL dbcsr_trace(n, m, numer)
                  CALL dbcsr_trace(grad, step, denom)
                  beta = numer/denom
               CASE (cg_fletcher_reeves)
                  CALL dbcsr_hadamard_product(prec, grad, n)
                  CALL dbcsr_trace(grad, n, denom)
                  CALL dbcsr_hadamard_product(prec, m, n)
                  CALL dbcsr_trace(m, n, numer)
                  beta = numer/denom
               CASE (cg_polak_ribiere)
                  CALL dbcsr_hadamard_product(prec, grad, n)
                  CALL dbcsr_trace(grad, n, denom)
                  CALL dbcsr_add(grad, m, -1.0_dp, 1.0_dp)
                  CALL dbcsr_hadamard_product(prec, grad, n)
                  CALL dbcsr_trace(n, m, numer)
                  beta = numer/denom
               CASE (cg_fletcher)
                  CALL dbcsr_hadamard_product(prec, m, n)
                  CALL dbcsr_trace(m, n, numer)
                  CALL dbcsr_trace(grad, step, denom)
                  beta = -1.0_dp*numer/denom
               CASE (cg_liu_storey)
                  CALL dbcsr_trace(grad, step, denom)
                  CALL dbcsr_add(grad, m, -1.0_dp, 1.0_dp)
                  CALL dbcsr_hadamard_product(prec, grad, n)
                  CALL dbcsr_trace(n, m, numer)
                  beta = -1.0_dp*numer/denom
               CASE (cg_dai_yuan)
                  CALL dbcsr_hadamard_product(prec, m, n)
                  CALL dbcsr_trace(m, n, numer)
                  CALL dbcsr_add(grad, m, -1.0_dp, 1.0_dp)
                  CALL dbcsr_trace(grad, step, denom)
                  beta = numer/denom
               CASE (cg_hager_zhang)
                  CALL dbcsr_add(grad, m, -1.0_dp, 1.0_dp)
                  CALL dbcsr_trace(grad, step, denom)
                  CALL dbcsr_hadamard_product(prec, grad, n)
                  CALL dbcsr_trace(n, grad, numer)
                  kappa = 2.0_dp*numer/denom
                  CALL dbcsr_trace(n, m, numer)
                  tau = numer/denom
                  CALL dbcsr_trace(step, m, numer)
                  beta = tau-kappa*numer/denom
               CASE (cg_zero)
                  beta = 0.0_dp
               CASE DEFAULT
                  CPABORT("illegal conjugator")
               END SELECT
            ENDIF ! iteration.eq.0

            ! move the current gradient to its storage
            CALL dbcsr_copy(grad, m)

            ! precondition new gradient (use m as tmp storage)
            CALL dbcsr_hadamard_product(prec, grad, m)
            CALL dbcsr_filter(m, eps_filter)

            ! recompute the step direction
            CALL dbcsr_add(step, m, beta, -1.0_dp)
            CALL dbcsr_filter(step, eps_filter)

!! ALTERNATIVE METHOD TO OBTAIN THE STEP FROM THE GRADIENT
!CALL dbcsr_init(qqqq)
!CALL dbcsr_create(qqqq,template=qq)
!CALL dbcsr_init(pppp)
!CALL dbcsr_create(pppp,template=pp)
!CALL dbcsr_init(zero_pq)
!CALL dbcsr_create(zero_pq,template=pq)
!CALL dbcsr_init(zero_qp)
!CALL dbcsr_create(zero_qp,template=qp)
!CALL dbcsr_multiply("T","N",1.0_dp,aux2,aux2,0.0_dp,qqqq,&
!        filter_eps=eps_filter)
!CALL dbcsr_multiply("N","T",-1.0_dp,aux1,aux1,0.0_dp,pppp,&
!        filter_eps=eps_filter)
!CALL dbcsr_set(zero_qp,0.0_dp)
!CALL dbcsr_set(zero_pq,0.0_dp)
!CALL solve_riccati_equation(pppp,qqqq,grad,zero_pq,zero_qp,zero_qp,&
!               .TRUE.,tensor_type,&
!               conjugator,max_iter,eps_convergence,eps_filter,&
!               converged,level+1)
!CALL dbcsr_release(qqqq)
!CALL dbcsr_release(pppp)
!CALL dbcsr_release(zero_qp)
!CALL dbcsr_release(zero_pq)

            ! calculate the optimal step size
            ! m=step.aux1+aux2.step
            IF (present_vv) THEN
               CALL dbcsr_multiply("N", "N", 1.0_dp, vv, step, 0.0_dp, vv_step, &
                                   filter_eps=eps_filter)
               CALL dbcsr_multiply("N", "N", 1.0_dp, vv_step, aux1, 0.0_dp, m, &
                                   filter_eps=eps_filter)
            ELSE
               CALL dbcsr_multiply("N", "N", 1.0_dp, step, aux1, 0.0_dp, m, &
                                   filter_eps=eps_filter)
            ENDIF
            IF (present_oo) THEN
               CALL dbcsr_multiply("N", "N", 1.0_dp, step, oo, 0.0_dp, step_oo, &
                                   filter_eps=eps_filter)
               CALL dbcsr_multiply("N", "N", 1.0_dp, aux2, step_oo, 1.0_dp, m, &
                                   filter_eps=eps_filter)
            ELSE
               CALL dbcsr_multiply("N", "N", 1.0_dp, aux2, step, 1.0_dp, m, &
                                   filter_eps=eps_filter)
            ENDIF

            IF (quadratic_term) THEN
               ! n=step.pq.step
               IF (present_oo) THEN
                  CALL dbcsr_multiply("N", "N", 1.0_dp, pq, step, 0.0_dp, oo1, &
                                      filter_eps=eps_filter)
                  CALL dbcsr_multiply("N", "N", 1.0_dp, oo1, oo, 0.0_dp, oo2, &
                                      filter_eps=eps_filter)
               ELSE
                  CALL dbcsr_multiply("N", "N", 1.0_dp, pq, step, 0.0_dp, oo2, &
                                      filter_eps=eps_filter)
               ENDIF
               IF (present_vv) THEN
                  CALL dbcsr_multiply("N", "N", 1.0_dp, step, oo2, 0.0_dp, res_trial, &
                                      filter_eps=eps_filter)
                  CALL dbcsr_multiply("N", "N", 1.0_dp, vv, res_trial, 0.0_dp, n, &
                                      filter_eps=eps_filter)
               ELSE
                  CALL dbcsr_multiply("N", "N", 1.0_dp, step, oo2, 0.0_dp, n, &
                                      filter_eps=eps_filter)
               ENDIF

            ELSE
               CALL dbcsr_set(n, 0.0_dp)
            ENDIF

            ! calculate coefficients of the cubic eq for alpha - step size
            c0 = 2.0_dp*(dbcsr_frobenius_norm(n))**2

            CALL dbcsr_trace(m, n, c1)
            c1 = -3.0_dp*c1

            CALL dbcsr_trace(res, n, c2)
            c2 = -2.0_dp*c2+(dbcsr_frobenius_norm(m))**2

            CALL dbcsr_trace(res, m, c3)

            ! find step size
            CALL analytic_line_search(c0, c1, c2, c3, step_size, nsteps)

            IF (nsteps .EQ. 0) THEN
               CPABORT("no step sizes!")
            ENDIF
            ! if we have several possible step sizes
            ! choose one with the lowest objective function
            best_norm = 1.0E+100_dp
            best_step_size = 0.0_dp
            DO istep = 1, nsteps
               ! recompute the residues
               CALL dbcsr_copy(res_trial, res)
               CALL dbcsr_add(res_trial, m, 1.0_dp, step_size(istep))
               CALL dbcsr_add(res_trial, n, 1.0_dp, -step_size(istep)*step_size(istep))
               CALL dbcsr_filter(res_trial, eps_filter)
               ! RZK-warning objective function might be different in the case of
               ! tensor_up_down
               !obj_function=0.5_dp*(dbcsr_frobenius_norm(res_trial))**2
               CALL dbcsr_norm(res_trial, dbcsr_norm_maxabsnorm, norm_scalar=obj_function)
               IF (obj_function .LT. best_norm) THEN
                  best_norm = obj_function
                  best_step_size = step_size(istep)
               ENDIF
            ENDDO

         ENDIF

         ! update X along the line
         CALL dbcsr_add(x, step, 1.0_dp, best_step_size)
         CALL dbcsr_filter(x, eps_filter)

         ! evaluate current energy correction
         !change_ecorr=ecorr
         !CALL dbcsr_trace(qp,x,ecorr,"T","N")
         !change_ecorr=ecorr-change_ecorr

         ! check convergence and other exit criteria
         converged = (best_norm .LT. eps_convergence)
         IF (converged .OR. (iteration .GE. max_iter)) THEN
            prepare_to_exit = .TRUE.
         ENDIF

         t2 = m_walltime()

         IF (unit_nr > 0) THEN
            WRITE (unit_nr, '(T6,A,1X,I4,1X,E12.3,F8.3)') &
               "RICCATI iter ", iteration, best_norm, t2-t1
            !WRITE(unit_nr,'(T6,A,1X,I4,1X,F15.9,F15.9,E12.3,F8.3)') &
            !   "RICCATI iter ",iteration,ecorr,change_ecorr,best_norm,t2-t1
         ENDIF

         t1 = m_walltime()

         iteration = iteration+1

         IF (prepare_to_exit) EXIT

      ENDDO

      CALL dbcsr_release(aux1)
      CALL dbcsr_release(aux2)
      CALL dbcsr_release(grad)
      CALL dbcsr_release(step)
      CALL dbcsr_release(n)
      CALL dbcsr_release(m)
      CALL dbcsr_release(oo1)
      CALL dbcsr_release(oo2)
      CALL dbcsr_release(res_trial)
      CALL dbcsr_release(vv_step)
      CALL dbcsr_release(step_oo)

      CALL timestop(handle)

   END SUBROUTINE solve_riccati_equation

! **************************************************************************************************
!> \brief Computes a preconditioner from diagonal elements of ~f_oo, ~f_vv
!>        The preconditioner is approximately equal to
!>        prec_ai ~ (e_a - e_i)^(-2)
!>        However, the real expression is more complex
!> \param prec ...
!> \param pp ...
!> \param qq ...
!> \param eps_filter ...
!> \par History
!>       2011.07 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
   SUBROUTINE create_preconditioner(prec, pp, qq, eps_filter)

      TYPE(dbcsr_type), INTENT(OUT)                      :: prec
      TYPE(dbcsr_type), INTENT(IN)                       :: pp, qq
      REAL(KIND=dp), INTENT(IN)                          :: eps_filter

      CHARACTER(len=*), PARAMETER :: routineN = 'create_preconditioner', &
         routineP = moduleN//':'//routineN

      INTEGER                                            :: col, handle, hold, iblock_col, &
                                                            iblock_row, mynode, nblkcols_tot, &
                                                            nblkrows_tot, p_nrows, q_nrows, row
      LOGICAL                                            :: tr
      REAL(KIND=dp), ALLOCATABLE, DIMENSION(:)           :: p_diagonal, q_diagonal
      REAL(KIND=dp), DIMENSION(:, :), POINTER            :: p_new_block
      TYPE(dbcsr_distribution_type)                      :: dist
      TYPE(dbcsr_type)                                   :: pp_diag, qq_diag, t1, t2, tmp

!LOGICAL, INTENT(IN)                      :: use_virt_orbs

      CALL timeset(routineN, handle)

!    ! copy diagonal elements
!    CALL dbcsr_get_info(pp,nfullrows_total=nrows)
!    CALL dbcsr_init(pp_diag)
!    CALL dbcsr_create(pp_diag,template=pp)
!    ALLOCATE(diagonal(nrows))
!    CALL dbcsr_get_diag(pp,diagonal)
!    CALL dbcsr_add_on_diag(pp_diag,1.0_dp)
!    CALL dbcsr_set_diag(pp_diag,diagonal)
!    DEALLOCATE(diagonal)
!
      ! initialize a matrix to 1.0
      CALL dbcsr_create(tmp, template=prec)
      ! use an ugly hack to set all elements of tmp to 1
      ! because dbcsr_set does not do it (despite its name)
      !CALL dbcsr_set(tmp,1.0_dp)
      CALL dbcsr_get_info(tmp, distribution=dist)
      CALL dbcsr_distribution_get(dist, mynode=mynode)
      CALL dbcsr_work_create(tmp, work_mutable=.TRUE.)
      nblkrows_tot = dbcsr_nblkrows_total(tmp)
      nblkcols_tot = dbcsr_nblkcols_total(tmp)
      DO row = 1, nblkrows_tot
         DO col = 1, nblkcols_tot
            tr = .FALSE.
            iblock_row = row
            iblock_col = col
            CALL dbcsr_get_stored_coordinates(tmp, iblock_row, iblock_col, hold)
            IF (hold .EQ. mynode) THEN
               NULLIFY (p_new_block)
               CALL dbcsr_reserve_block2d(tmp, iblock_row, iblock_col, p_new_block)
               CPASSERT(ASSOCIATED(p_new_block))
               p_new_block(:, :) = 1.0_dp
            ENDIF ! mynode
         ENDDO
      ENDDO
      CALL dbcsr_finalize(tmp)

      ! copy diagonal elements of pp into cols of a matrix
      CALL dbcsr_get_info(pp, nfullrows_total=p_nrows)
      CALL dbcsr_create(pp_diag, template=pp)
      ALLOCATE (p_diagonal(p_nrows))
      CALL dbcsr_get_diag(pp, p_diagonal)
      CALL dbcsr_add_on_diag(pp_diag, 1.0_dp)
      CALL dbcsr_set_diag(pp_diag, p_diagonal)
      ! RZK-warning is it possible to use dbcsr_scale_by_vector?
      ! or even insert elements directly in the prev cycles
      CALL dbcsr_create(t2, template=prec)
      CALL dbcsr_multiply("N", "N", 1.0_dp, tmp, pp_diag, &
                          0.0_dp, t2, filter_eps=eps_filter)

      ! copy diagonal elements qq into rows of a matrix
      CALL dbcsr_get_info(qq, nfullrows_total=q_nrows)
      CALL dbcsr_create(qq_diag, template=qq)
      ALLOCATE (q_diagonal(q_nrows))
      CALL dbcsr_get_diag(qq, q_diagonal)
      CALL dbcsr_add_on_diag(qq_diag, 1.0_dp)
      CALL dbcsr_set_diag(qq_diag, q_diagonal)
      CALL dbcsr_set(tmp, 1.0_dp)
      CALL dbcsr_create(t1, template=prec)
      CALL dbcsr_multiply("N", "N", 1.0_dp, qq_diag, tmp, &
                          0.0_dp, t1, filter_eps=eps_filter)

      CALL dbcsr_hadamard_product(t1, t2, prec)
      CALL dbcsr_release(t1)
      CALL dbcsr_scale(prec, 2.0_dp)

      ! Get the diagonal of tr(qq).qq
      CALL dbcsr_multiply("T", "N", 1.0_dp, qq, qq, &
                          0.0_dp, qq_diag, retain_sparsity=.TRUE., &
                          filter_eps=eps_filter)
      CALL dbcsr_get_diag(qq_diag, q_diagonal)
      CALL dbcsr_set(qq_diag, 0.0_dp)
      CALL dbcsr_add_on_diag(qq_diag, 1.0_dp)
      CALL dbcsr_set_diag(qq_diag, q_diagonal)
      DEALLOCATE (q_diagonal)
      CALL dbcsr_set(tmp, 1.0_dp)
      CALL dbcsr_multiply("N", "N", 1.0_dp, qq_diag, tmp, &
                          0.0_dp, t2, filter_eps=eps_filter)
      CALL dbcsr_release(qq_diag)
      CALL dbcsr_add(prec, t2, 1.0_dp, 1.0_dp)

      ! Get the diagonal of pp.tr(pp)
      CALL dbcsr_multiply("N", "T", 1.0_dp, pp, pp, &
                          0.0_dp, pp_diag, retain_sparsity=.TRUE., &
                          filter_eps=eps_filter)
      CALL dbcsr_get_diag(pp_diag, p_diagonal)
      CALL dbcsr_set(pp_diag, 0.0_dp)
      CALL dbcsr_add_on_diag(pp_diag, 1.0_dp)
      CALL dbcsr_set_diag(pp_diag, p_diagonal)
      DEALLOCATE (p_diagonal)
      CALL dbcsr_set(tmp, 1.0_dp)
      CALL dbcsr_multiply("N", "N", 1.0_dp, tmp, pp_diag, &
                          0.0_dp, t2, filter_eps=eps_filter)
      CALL dbcsr_release(tmp)
      CALL dbcsr_release(pp_diag)
      CALL dbcsr_add(prec, t2, 1.0_dp, 1.0_dp)

      ! now add the residual component
      !CALL dbcsr_hadamard_product(res,qp,t2)
      !CALL dbcsr_add(prec,t2,1.0_dp,-2.0_dp)
      CALL dbcsr_release(t2)
      CALL dbcsr_function_of_elements(prec, func=dbcsr_func_inverse)
      CALL dbcsr_filter(prec, eps_filter)

      CALL timestop(handle)

   END SUBROUTINE create_preconditioner

! **************************************************************************************************
!> \brief Finds real roots of a cubic equation
!>    >        a*x**3 + b*x**2 + c*x + d = 0
!>        and returns only those roots for which the derivative is positive
!>
!>   Step 0: Check the true order of the equation. Cubic, quadratic, linear?
!>   Step 1: Calculate p and q
!>           p = ( 3*c/a - (b/a)**2 ) / 3
!>           q = ( 2*(b/a)**3 - 9*b*c/a/a + 27*d/a ) / 27
!>   Step 2: Calculate discriminant D
!>           D = (p/3)**3 + (q/2)**2
!>   Step 3: Depending on the sign of D, we follow different strategy.
!>           If D<0, three distinct real roots.
!>           If D=0, three real roots of which at least two are equal.
!>           If D>0, one real and two complex roots.
!>   Step 3a: For D>0 and D=0,
!>           Calculate u and v
!>           u = cubic_root(-q/2 + sqrt(D))
!>           v = cubic_root(-q/2 - sqrt(D))
!>           Find the three transformed roots
!>           y1 = u + v
!>           y2 = -(u+v)/2 + i (u-v)*sqrt(3)/2
!>           y3 = -(u+v)/2 - i (u-v)*sqrt(3)/2
!>   Step 3b Alternately, for D<0, a trigonometric formulation is more convenient
!>           y1 =  2 * sqrt(|p|/3) * cos(phi/3)
!>           y2 = -2 * sqrt(|p|/3) * cos((phi+pi)/3)
!>           y3 = -2 * sqrt(|p|/3) * cos((phi-pi)/3)
!>           where phi = acos(-q/2/sqrt(|p|**3/27))
!>                 pi  = 3.141592654...
!>   Step 4  Find the real roots
!>           x = y - b/a/3
!>   Step 5  Check the derivative and return only those real roots
!>           for which the derivative is positive
!>
!> \param a ...
!> \param b ...
!> \param c ...
!> \param d ...
!> \param minima ...
!> \param nmins ...
!> \par History
!>       2011.06 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
   SUBROUTINE analytic_line_search(a, b, c, d, minima, nmins)

      REAL(KIND=dp), INTENT(IN)                          :: a, b, c, d
      REAL(KIND=dp), DIMENSION(3), INTENT(OUT)           :: minima
      INTEGER, INTENT(OUT)                               :: nmins

      CHARACTER(len=*), PARAMETER :: routineN = 'analytic_line_search', &
         routineP = moduleN//':'//routineN

      INTEGER                                            :: i, nroots
      REAL(KIND=dp)                                      :: DD, der, p, phi, pi, q, temp1, temp2, u, &
                                                            v, y1, y2, y2i, y2r, y3
      REAL(KIND=dp), DIMENSION(3)                        :: x

!    CALL timeset(routineN,handle)

      pi = ACOS(-1.0_dp)

      ! Step 0: Check coefficients and find the true order of the eq
      IF (a .EQ. 0.0_dp) THEN
         IF (b .EQ. 0.0_dp) THEN
            IF (c .EQ. 0.0_dp) THEN
               ! Non-equation, no valid solutions
               nroots = 0
            ELSE
               ! Linear equation with one root.
               nroots = 1
               x(1) = -d/c
            ENDIF
         ELSE
            ! Quadratic equation with max two roots.
            DD = c*c-4.0_dp*b*d
            IF (DD .GT. 0.0_dp) THEN
               nroots = 2
               x(1) = (-c+SQRT(DD))/2.0_dp/b
               x(2) = (-c-SQRT(DD))/2.0_dp/b
            ELSE IF (DD .LT. 0.0_dp) THEN
               nroots = 0
            ELSE
               nroots = 1
               x(1) = -c/2.0_dp/b
            ENDIF
         ENDIF
      ELSE
         ! Cubic equation with max three roots
         ! Calculate p and q
         p = c/a-b*b/a/a/3.0_dp
         q = (2.0_dp*b*b*b/a/a/a-9.0_dp*b*c/a/a+27.0_dp*d/a)/27.0_dp

         ! Calculate DD
         DD = p*p*p/27.0_dp+q*q/4.0_dp

         IF (DD .LT. 0.0_dp) THEN
            ! three real unequal roots -- use the trigonometric formulation
            phi = ACOS(-q/2.0_dp/SQRT(ABS(p*p*p)/27.0_dp))
            temp1 = 2.0_dp*SQRT(ABS(p)/3.0_dp)
            y1 = temp1*COS(phi/3.0_dp)
            y2 = -temp1*COS((phi+pi)/3.0_dp)
            y3 = -temp1*COS((phi-pi)/3.0_dp)
         ELSE
            ! 1 real & 2 conjugate complex roots OR 3 real roots (some are equal)
            temp1 = -q/2.0_dp+SQRT(DD)
            temp2 = -q/2.0_dp-SQRT(DD)
            u = ABS(temp1)**(1.0_dp/3.0_dp)
            v = ABS(temp2)**(1.0_dp/3.0_dp)
            IF (temp1 .LT. 0.0_dp) u = -u
            IF (temp2 .LT. 0.0_dp) v = -v
            y1 = u+v
            y2r = -(u+v)/2.0_dp
            y2i = (u-v)*SQRT(3.0_dp)/2.0_dp
         ENDIF

         ! Final transformation
         temp1 = b/a/3.0_dp
         y1 = y1-temp1
         y2 = y2-temp1
         y3 = y3-temp1
         y2r = y2r-temp1

         ! Assign answers
         IF (DD .LT. 0.0_dp) THEN
            nroots = 3
            x(1) = y1
            x(2) = y2
            x(3) = y3
         ELSE IF (DD .EQ. 0.0_dp) THEN
            nroots = 2
            x(1) = y1
            x(2) = y2r
            !x(3) = cmplx(y2r,  0.)
         ELSE
            nroots = 1
            x(1) = y1
            !x(2) = cmplx(y2r, y2i)
            !x(3) = cmplx(y2r,-y2i)
         ENDIF

      ENDIF

!write(*,'(i2,a)') nroots, ' real root(s)'
      nmins = 0
      DO i = 1, nroots
         ! maximum or minimum? use the derivative
         ! 3*a*x**2+2*b*x+c
         der = 3.0_dp*a*x(i)*x(i)+2.0_dp*b*x(i)+c
         IF (der .GT. 0.0_dp) THEN
            nmins = nmins+1
            minima(nmins) = x(i)
!write(*,'(a,i2,a,f10.5)') 'Minimum ', i, ', value: ', x(i)
         ENDIF
      ENDDO

!    CALL timestop(handle)

   END SUBROUTINE analytic_line_search

! **************************************************************************************************
!> \brief Diagonalizes diagonal blocks of a symmetric dbcsr matrix
!>        and returs its eigenvectors
!> \param matrix ...
!> \param c ...
!> \param e ...
!> \par History
!>       2011.07 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
   SUBROUTINE diagonalize_diagonal_blocks(matrix, c, e)

      TYPE(dbcsr_type), INTENT(IN)                       :: matrix
      TYPE(dbcsr_type), INTENT(OUT)                      :: c
      TYPE(dbcsr_type), INTENT(OUT), OPTIONAL            :: e

      CHARACTER(len=*), PARAMETER :: routineN = 'diagonalize_diagonal_blocks', &
         routineP = moduleN//':'//routineN

      INTEGER                                            :: handle, iblock_col, iblock_row, &
                                                            iblock_size, info, lwork, orbital
      LOGICAL                                            :: block_needed, do_eigenvalues
      REAL(kind=dp), ALLOCATABLE, DIMENSION(:)           :: eigenvalues, work
      REAL(kind=dp), ALLOCATABLE, DIMENSION(:, :)        :: data_copy
      REAL(kind=dp), DIMENSION(:, :), POINTER            :: data_p, p_new_block
      TYPE(dbcsr_iterator_type)                          :: iter

      CALL timeset(routineN, handle)

      IF (PRESENT(e)) THEN
         do_eigenvalues = .TRUE.
      ELSE
         do_eigenvalues = .FALSE.
      ENDIF

      ! create a matrix for eigenvectors
      CALL dbcsr_work_create(c, work_mutable=.TRUE.)
      IF (do_eigenvalues) &
         CALL dbcsr_work_create(e, work_mutable=.TRUE.)

      CALL dbcsr_iterator_start(iter, matrix)

      DO WHILE (dbcsr_iterator_blocks_left(iter))

         CALL dbcsr_iterator_next_block(iter, iblock_row, iblock_col, data_p, row_size=iblock_size)

         block_needed = .FALSE.
         IF (iblock_row == iblock_col) block_needed = .TRUE.

         IF (block_needed) THEN

            ! Prepare data
            ALLOCATE (eigenvalues(iblock_size))
            ALLOCATE (data_copy(iblock_size, iblock_size))
            data_copy(:, :) = data_p(:, :)

            ! Query the optimal workspace for dsyev
            LWORK = -1
            ALLOCATE (WORK(MAX(1, LWORK)))
            CALL DSYEV('V', 'L', iblock_size, data_copy, iblock_size, eigenvalues, WORK, LWORK, INFO)
            LWORK = INT(WORK(1))
            DEALLOCATE (WORK)

            ! Allocate the workspace and solve the eigenproblem
            ALLOCATE (WORK(MAX(1, LWORK)))
            CALL DSYEV('V', 'L', iblock_size, data_copy, iblock_size, eigenvalues, WORK, LWORK, INFO)
            IF (INFO .NE. 0) THEN
               CPABORT("DSYEV failed")
            END IF

            ! copy eigenvectors into a cp_dbcsr matrix
            NULLIFY (p_new_block)
            CALL dbcsr_reserve_block2d(c, iblock_row, iblock_col, p_new_block)
            CPASSERT(ASSOCIATED(p_new_block))
            p_new_block(:, :) = data_copy(:, :)

            ! if requested copy eigenvalues into a cp_dbcsr matrix
            IF (do_eigenvalues) THEN
               NULLIFY (p_new_block)
               CALL dbcsr_reserve_block2d(e, iblock_row, iblock_col, p_new_block)
               CPASSERT(ASSOCIATED(p_new_block))
               p_new_block(:, :) = 0.0_dp
               DO orbital = 1, iblock_size
                  p_new_block(orbital, orbital) = eigenvalues(orbital)
               ENDDO
            ENDIF

            DEALLOCATE (WORK)
            DEALLOCATE (data_copy)
            DEALLOCATE (eigenvalues)

         ENDIF

      ENDDO

      CALL dbcsr_iterator_stop(iter)

      CALL dbcsr_finalize(c)
      IF (do_eigenvalues) CALL dbcsr_finalize(e)

      CALL timestop(handle)

   END SUBROUTINE diagonalize_diagonal_blocks

! **************************************************************************************************
!> \brief Transforms a matrix M_out = tr(U1) * M_in * U2
!> \param matrix ...
!> \param u1 ...
!> \param u2 ...
!> \param eps_filter ...
!> \par History
!>       2011.10 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
   SUBROUTINE matrix_forward_transform(matrix, u1, u2, eps_filter)

      TYPE(dbcsr_type), INTENT(INOUT)                    :: matrix
      TYPE(dbcsr_type), INTENT(IN)                       :: u1, u2
      REAL(KIND=dp), INTENT(IN)                          :: eps_filter

      CHARACTER(len=*), PARAMETER :: routineN = 'matrix_forward_transform', &
         routineP = moduleN//':'//routineN

      INTEGER                                            :: handle
      TYPE(dbcsr_type)                                   :: tmp

      CALL timeset(routineN, handle)

      CALL dbcsr_create(tmp, template=matrix, &
                        matrix_type=dbcsr_type_no_symmetry)
      CALL dbcsr_multiply("N", "N", 1.0_dp, matrix, u2, 0.0_dp, tmp, &
                          filter_eps=eps_filter)
      CALL dbcsr_multiply("T", "N", 1.0_dp, u1, tmp, 0.0_dp, matrix, &
                          filter_eps=eps_filter)
      CALL dbcsr_release(tmp)

      CALL timestop(handle)

   END SUBROUTINE matrix_forward_transform

! **************************************************************************************************
!> \brief Transforms a matrix M_out = U1 * M_in * tr(U2)
!> \param matrix ...
!> \param u1 ...
!> \param u2 ...
!> \param eps_filter ...
!> \par History
!>       2011.10 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
   SUBROUTINE matrix_backward_transform(matrix, u1, u2, eps_filter)

      TYPE(dbcsr_type), INTENT(INOUT)                    :: matrix
      TYPE(dbcsr_type), INTENT(IN)                       :: u1, u2
      REAL(KIND=dp), INTENT(IN)                          :: eps_filter

      CHARACTER(len=*), PARAMETER :: routineN = 'matrix_backward_transform', &
         routineP = moduleN//':'//routineN

      INTEGER                                            :: handle
      TYPE(dbcsr_type)                                   :: tmp

      CALL timeset(routineN, handle)

      CALL dbcsr_create(tmp, template=matrix, &
                        matrix_type=dbcsr_type_no_symmetry)
      CALL dbcsr_multiply("N", "T", 1.0_dp, matrix, u2, 0.0_dp, tmp, &
                          filter_eps=eps_filter)
      CALL dbcsr_multiply("N", "N", 1.0_dp, u1, tmp, 0.0_dp, matrix, &
                          filter_eps=eps_filter)
      CALL dbcsr_release(tmp)

      CALL timestop(handle)

   END SUBROUTINE matrix_backward_transform

!! **************************************************************************************************
!!> \brief Transforms to a representation in which diagonal blocks
!!>        of qq and pp matrices are diagonal. This can improve convergence
!!>        of PCG
!!> \par History
!!>       2011.07 created [Rustam Z Khaliullin]
!!> \author Rustam Z Khaliullin
!! **************************************************************************************************
!  SUBROUTINE transform_matrices_to_blk_diag(matrix_pp,matrix_qq,matrix_qp,&
!    matrix_pq,eps_filter)
!
!    TYPE(dbcsr_type), INTENT(INOUT)       :: matrix_pp, matrix_qq,&
!                                                matrix_qp, matrix_pq
!    REAL(KIND=dp), INTENT(IN)                :: eps_filter
!
!    CHARACTER(len=*), PARAMETER :: routineN = 'transform_matrices_to_blk_diag',&
!      routineP = moduleN//':'//routineN
!
!    TYPE(dbcsr_type)                      :: tmp_pp, tmp_qq,&
!                                                tmp_qp, tmp_pq,&
!                                                blk, blk2
!    INTEGER                                  :: handle
!
!    CALL timeset(routineN,handle)
!
!    ! find a better basis by diagonalizing diagonal blocks
!    ! first pp
!    CALL dbcsr_init(blk)
!    CALL dbcsr_create(blk,template=matrix_pp)
!    CALL diagonalize_diagonal_blocks(matrix_pp,blk)
!
!    ! convert matrices to the new basis
!    CALL dbcsr_init(tmp_pp)
!    CALL dbcsr_create(tmp_pp,template=matrix_pp)
!    CALL dbcsr_multiply("N","N",1.0_dp,matrix_pp,blk,0.0_dp,tmp_pp,&
!               filter_eps=eps_filter)
!    CALL dbcsr_multiply("T","N",1.0_dp,blk,tmp_pp,0.0_dp,matrix_pp,&
!               filter_eps=eps_filter)
!    CALL dbcsr_release(tmp_pp)
!
!    ! now qq
!    CALL dbcsr_init(blk2)
!    CALL dbcsr_create(blk2,template=matrix_qq)
!    CALL diagonalize_diagonal_blocks(matrix_qq,blk2)
!
!    CALL dbcsr_init(tmp_qq)
!    CALL dbcsr_create(tmp_qq,template=matrix_qq)
!    CALL dbcsr_multiply("N","N",1.0_dp,matrix_qq,blk2,0.0_dp,tmp_qq,&
!               filter_eps=eps_filter)
!    CALL dbcsr_multiply("T","N",1.0_dp,blk2,tmp_qq,0.0_dp,matrix_qq,&
!               filter_eps=eps_filter)
!    CALL dbcsr_release(tmp_qq)
!
!    ! transform pq
!    CALL dbcsr_init(tmp_pq)
!    CALL dbcsr_create(tmp_pq,template=matrix_pq)
!    CALL dbcsr_multiply("T","N",1.0_dp,blk,matrix_pq,0.0_dp,tmp_pq,&
!               filter_eps=eps_filter)
!    CALL dbcsr_multiply("N","N",1.0_dp,tmp_pq,blk2,0.0_dp,matrix_pq,&
!               filter_eps=eps_filter)
!    CALL dbcsr_release(tmp_pq)
!
!    ! transform qp
!    CALL dbcsr_init(tmp_qp)
!    CALL dbcsr_create(tmp_qp,template=matrix_qp)
!    CALL dbcsr_multiply("N","N",1.0_dp,matrix_qp,blk,0.0_dp,tmp_qp,&
!               filter_eps=eps_filter)
!    CALL dbcsr_multiply("T","N",1.0_dp,blk2,tmp_qp,0.0_dp,matrix_qp,&
!               filter_eps=eps_filter)
!    CALL dbcsr_release(tmp_qp)
!
!    CALL dbcsr_release(blk2)
!    CALL dbcsr_release(blk)
!
!    CALL timestop(handle)
!
!  END SUBROUTINE transform_matrices_to_blk_diag

! **************************************************************************************************
!> \brief computes oo, ov, vo, and vv blocks of the ks matrix
!> \par History
!>       2011.06 created [Rustam Z Khaliullin]
!> \author Rustam Z Khaliullin
! **************************************************************************************************
!  SUBROUTINE ct_step_env_execute(env)
!
!    TYPE(ct_step_env_type)                      :: env
!
!    CHARACTER(len=*), PARAMETER :: routineN = 'ct_step_env_execute', &
!      routineP = moduleN//':'//routineN
!
!    INTEGER                                  :: handle
!
!    CALL timeset(routineN,handle)
!
!
!    CALL timestop(handle)
!
!  END SUBROUTINE ct_step_env_execute

END MODULE ct_methods

