fft.F90

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!! Copyright (C) 2002-2006 M. Marques, A. Castro, A. Rubio, G. Bertsch
!! Copyright (C) 2011 J. Alberdi-Rodriguez, P. Garcia Risueño, M. Oliveira
!!
!! This program is free software; you can redistribute it and/or modify
!! it under the terms of the GNU General Public License as published by
!! the Free Software Foundation; either version 2, or (at your option)
!! any later version.
!!
!! This program is distributed in the hope that it will be useful,
!! but WITHOUT ANY WARRANTY; without even the implied warranty of
!! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
!! GNU General Public License for more details.
!!
!! You should have received a copy of the GNU General Public License
!! along with this program; if not, write to the Free Software
!! Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA
!! 02110-1301, USA.
!!
 
#include "global.h"
 
module fft_oct_m
  use, intrinsic :: iso_c_binding
 
  use accel_oct_m
#ifdef HAVE_OPENCL
  use cl
#ifdef HAVE_CLFFT
  use clfft
#endif
#endif
  use fftw_oct_m
  use fftw_params_oct_m
  use debug_oct_m
  use global_oct_m
  use, intrinsic :: iso_fortran_env
  use lattice_vectors_oct_m
  use lalg_basic_oct_m
  use loct_math_oct_m
  use messages_oct_m
  use mpi_oct_m
  use namespace_oct_m
#ifdef HAVE_NFFT
  use nfft_oct_m
#endif
#if defined(HAVE_OPENMP) && defined(HAVE_FFTW3_THREADS)
  use omp_lib
#endif
  use parser_oct_m
  use pfft_oct_m
  use pfft_params_oct_m
  use pnfft_oct_m
  use profiling_oct_m
  use types_oct_m
  use unit_system_oct_m
  use varinfo_oct_m
 
  implicit none
 
  private
  public ::            &
    fft_t,             &
    fft_all_init,      &
    fft_all_end,       &
    fft_init,          &
    fft_init_stage1,   &
    fft_end,           &
    fft_copy,          &
    fft_get_dims,      &
    pad_feq,           &
    dfft_forward,      &
    zfft_forward,      &
    dfft_backward,     &
    zfft_backward,     &
    fft_scaling_factor,&
    fft_gg_transform,  &
    fft_get_ecut_from_box
 
 
  integer, public, parameter :: &
    FFT_NONE    = 0,         &
    fft_real    = 1,         &
    fft_complex = 2
 
  integer, public, parameter :: &
    FFTLIB_NONE  = 0, &
    fftlib_fftw  = 1, &
    fftlib_pfft  = 2, &
    fftlib_accel = 3, &
    fftlib_nfft  = 4, &
    fftlib_pnfft = 5
 
  integer, parameter :: &
    FFT_MAX  = 10, &
    fft_null = -1
 
  type fft_t
    private
    integer         :: slot = 0
 
    integer, public :: type
    integer, public :: library
 
    type(MPI_Comm)  :: comm
    integer         :: rs_n_global(3)
    integer         :: fs_n_global(3)
    integer         :: rs_n(3)
    integer         :: fs_n(3)
    integer         :: rs_istart(1:3)
    integer         :: fs_istart(1:3)
 
    integer, public :: stride_rs(1:3)
    integer, public :: stride_fs(1:3)
 
    type(c_ptr) :: planf
    type(c_ptr) :: planb
    !integer(ptrdiff_t_kind) :: pfft_planf !< PFFT plan for forward transform
    !integer(ptrdiff_t_kind) :: pfft_planb !< PFFT plan for backward transform
 
    real(real64), pointer, public :: drs_data(:,:,:)
    complex(real64), pointer, public :: zrs_data(:,:,:)
    complex(real64), pointer, public ::  fs_data(:,:,:)
#ifdef HAVE_CLFFT
 
    type(clfftPlanHandle) :: cl_plan_fw
    type(clfftPlanHandle) :: cl_plan_bw
#endif
    type(c_ptr)           :: cuda_plan_fw
    type(c_ptr)           :: cuda_plan_bw
#ifdef HAVE_NFFT
    type(nfft_t),  public :: nfft
#endif
    type(pnfft_t), public :: pnfft
 
    logical, public :: aligned_memory
  end type fft_t
 
  interface dfft_forward
    module procedure dfft_forward_1d, dfft_forward_accel, dfft_forward_3d
  end interface dfft_forward
 
  interface zfft_forward
    module procedure zfft_forward_1d, zfft_forward_accel, zfft_forward_3d
  end interface zfft_forward
 
  interface dfft_backward
    module procedure dfft_backward_1d, dfft_backward_accel, dfft_backward_3d
  end interface dfft_backward
 
  interface zfft_backward
    module procedure zfft_backward_1d, zfft_backward_accel, zfft_backward_3d
  end interface zfft_backward
 
  logical, save, public :: fft_initialized = .false.
  integer, save         :: fft_refs(FFT_MAX)
  type(fft_t), save     :: fft_array(FFT_MAX)
  logical               :: fft_optimize
  integer, save         :: fft_prepare_plan
  integer, public       :: fft_default_lib = -1
#ifdef HAVE_NFFT
  type(nfft_t), save    :: nfft_options
#endif
  type(pnfft_t), save   :: pnfft_options
 
  integer, parameter ::  &
    CUFFT_R2C = int(z'2a'),   &
    cufft_c2r = int(z'2c'),   &
    cufft_c2c = int(z'29'),   &
    cufft_d2z = int(z'6a'),   &
    cufft_z2d = int(z'6c'),   &
    cufft_z2z = int(z'69')
 
contains
 
  ! ---------------------------------------------------------
  subroutine fft_all_init(namespace)
    type(namespace_t),      intent(in)   :: namespace
 
    integer :: ii, fft_default
#if defined(HAVE_OPENMP) && defined(HAVE_FFTW3_THREADS)
    integer :: iret
#endif
 
    push_sub(fft_all_init)
 
    fft_initialized = .true.
 
    !%Variable FFTOptimize
    !%Type logical
    !%Default yes
    !%Section Mesh::FFTs
    !%Description
    !% Should <tt>octopus</tt> optimize the FFT dimensions?
    !% This means that the mesh to which FFTs are applied is not taken to be as small
    !% as possible: some points may be added to each direction in order to get a "good number"
    !% for the performance of the FFT algorithm.
    !% The best FFT grid dimensions are given by <math>2^a 3^b 5^c 7^d 11^e 13^f</math>
    !% where <math>a,b,c,d</math> are arbitrary and <math>e,f</math> are 0 or 1.
    !% (<a href=http://www.fftw.org/doc/Complex-DFTs.html>ref</a>).
    !% In some cases, namely when using
    !% the split-operator, or Suzuki-Trotter propagators, this option should be turned off.
    !% For spatial FFTs in periodic directions, the grid is never optimized, but a warning will
    !% be written if the number is not good, with a suggestion of a better one to use, so you
    !% can try a different spacing if you want to get a good number.
    !%End
    call parse_variable(namespace, 'FFTOptimize', .true., fft_optimize)
    do ii = 1, fft_max
      fft_refs(ii) = fft_null
    end do
 
    !%Variable FFTPreparePlan
    !%Type integer
    !%Default fftw_measure
    !%Section Mesh::FFTs
    !%Description
    !% The FFTs are performed in octopus with the help of <a href=http://www.fftw.org>FFTW</a> and similar packages.
    !% Before doing the actual computations, this package prepares a "plan", which means that
    !% the precise numerical strategy to be followed to compute the FFT is machine/compiler-dependent,
    !% and therefore the software attempts to figure out which is this precise strategy (see the
    !% FFTW documentation for details). This plan preparation, which has to be done for each particular
    !% FFT shape, can be done exhaustively and carefully (slow), or merely estimated. Since this is
    !% a rather critical numerical step, by default it is done carefully, which implies a longer initial
    !% initialization, but faster subsequent computations. You can change this behaviour by changing
    !% this <tt>FFTPreparePlan</tt> variable, and in this way you can force FFTW to do a fast guess or
    !% estimation of which is the best way to perform the FFT.
    !%Option fftw_measure 0
    !% This plan implies a longer initialization, but involves a more careful analysis
    !% of the strategy to follow, and therefore more efficient FFTs. A side effect of the runtime
    !% choices is that this plan can introduce slight numerical fluctuations between runs.
    !%Option fftw_estimate 64
    !% This is the "fast initialization" scheme, in which the plan is merely guessed from "reasonable"
    !% assumptions. This is the default option, as it guarantees stable results
    !%Option fftw_patient 32
    !% It is like fftw_measure, but considers a wider range of algorithms and often produces a
    !% "more optimal" plan (especially for large transforms), but at the expense of several times
    !% longer planning time (especially for large transforms).
    !%Option fftw_exhaustive 8
    !% It is like fftw_patient, but considers an even wider range of algorithms,
    !% including many that we think are unlikely to be fast, to produce the most optimal
    !%  plan but with a substantially increased planning time.
    !%End
    call parse_variable(namespace, 'FFTPreparePlan', fftw_estimate, fft_prepare_plan)
    if (.not. varinfo_valid_option('FFTPreparePlan', fft_prepare_plan)) then
      call messages_input_error(namespace, 'FFTPreparePlan')
    end if
 
    !%Variable FFTLibrary
    !%Type integer
    !%Section Mesh::FFTs
    !%Default fftw
    !%Description
    !% (experimental) You can select the FFT library to use.
    !%Option fftw 1
    !% Uses FFTW3 library.
    !%Option pfft 2
    !% (experimental) Uses PFFT library, which has to be linked.
    !%Option accel 3
    !% (experimental) Uses a GPU accelerated library. This only
    !% works if Octopus was compiled with Cuda or OpenCL support.
    !%End
    fft_default = fftlib_fftw
    if(accel_is_enabled()) then
      fft_default = fftlib_accel
    end if
    call parse_variable(namespace, 'FFTLibrary', fft_default, fft_default_lib)
 
    if (fft_default_lib == fftlib_accel) then
#if ! (defined(HAVE_CLFFT) || defined(HAVE_CUDA))
      call messages_write('You have selected the Accelerated FFT, but Octopus was compiled', new_line = .true.)
      call messages_write('without clfft (OpenCL) or Cuda support.')
      call messages_fatal()
#endif
      if (.not. accel_is_enabled()) then
        call messages_write('You have selected the accelerated FFT, but acceleration is disabled.')
        call messages_fatal()
      end if
    end if
 
#if defined(HAVE_OPENMP) && defined(HAVE_FFTW3_THREADS)
    if (omp_get_max_threads() > 1) then
 
      call messages_write('Info: Initializing Multi-threaded FFTW')
      call messages_info()
 
      iret = fftw_init_threads()
      if (iret == 0) then
        call messages_write('Initialization of FFTW3 threads failed.')
        call messages_fatal()
      end if
      call fftw_plan_with_nthreads(omp_get_max_threads())
 
    end if
#endif
#ifdef HAVE_NFFT
    call nfft_guru_options(nfft_options, namespace)
#endif
    call pnfft_guru_options(pnfft_options, namespace)
 
    pop_sub(fft_all_init)
  end subroutine fft_all_init
 
 
  ! ---------------------------------------------------------
  subroutine fft_all_end()
    integer :: ii
 
    push_sub(fft_all_end)
 
    do ii = 1, fft_max
      if (fft_refs(ii) /= fft_null) then
        call fft_end(fft_array(ii))
      end if
    end do
 
#ifdef HAVE_PFFT
    call pfft_cleanup()
#endif
 
#if defined(HAVE_OPENMP) && defined(HAVE_FFTW3_THREADS)
    call fftw_cleanup_threads()
#else
    call fftw_cleanup()
#endif
 
    fft_initialized = .false.
 
    pop_sub(fft_all_end)
  end subroutine fft_all_end
 
  ! ---------------------------------------------------------
  subroutine fft_init(this, nn, dim, type, library, optimize, optimize_parity, comm, mpi_grp, use_aligned)
    type(fft_t),       intent(inout) :: this
    integer,           intent(inout) :: nn(3)
    integer,           intent(in)    :: dim
    integer,           intent(in)    :: type
    integer,           intent(in)    :: library
    logical,           intent(in)    :: optimize(3)
    integer,           intent(in)    :: optimize_parity(3)
    type(mpi_comm), optional, intent(out) :: comm
    type(mpi_grp_t), optional, intent(in) :: mpi_grp
    logical, optional                :: use_aligned
 
    integer :: ii, jj, fft_dim, idir, column_size, row_size, n3
    integer :: n_1, n_2, n_3, nn_temp(3)
    integer :: library_
    type(mpi_grp_t) :: mpi_grp_
    integer(int64) :: number_points, alloc_size
 
#ifdef HAVE_CLFFT
    real(real64) :: scale
    integer :: status
#endif
#ifdef HAVE_PFFT
    integer :: ierror
#endif
 
    push_sub(fft_init)
 
    assert(fft_initialized)
 
    assert(type == fft_real .or. type == fft_complex)
 
    mpi_grp_ = mpi_world
    if (present(mpi_grp)) mpi_grp_ = mpi_grp
 
    this%aligned_memory = optional_default(use_aligned, .false.)
 
    ! First, figure out the dimensionality of the FFT.
    fft_dim = 0
    do ii = 1, dim
      if (nn(ii) <= 1) exit
      fft_dim = fft_dim + 1
    end do
 
    if (fft_dim == 0) then
      message(1) = "Internal error in fft_init: apparently, a 1x1x1 FFT is required."
      call messages_fatal(1)
    end if
 
    if (fft_dim > 3) call messages_not_implemented('FFT for dimension > 3')
 
    library_ = library
    nn_temp(1:fft_dim) = nn(1:fft_dim)
 
    select case (library_)
    case (fftlib_accel)
      ! FFT optimization
      if(any(optimize_parity(1:fft_dim) > 1)) then
        message(1) = "Internal error in fft_init: optimize_parity must be negative, 0, or 1."
        call messages_fatal(1)
      end if
 
      do ii = 1, fft_dim
        nn_temp(ii) = fft_size(nn(ii), (/2, 3, 5, 7/), optimize_parity(ii))
        if (fft_optimize .and. optimize(ii)) nn(ii) = nn_temp(ii)
      end do
 
      ! if we can't optimize, in some cases we can't use the library
      if (any(nn(1:fft_dim) /= nn_temp(1:fft_dim))) then
#ifdef HAVE_CLFFT
        call messages_write('Invalid grid size for accel fft. FFTW will be used instead.')
        call messages_warning()
        library_ = fftlib_fftw
#endif
      end if
 
    case (fftlib_nfft)
 
      do ii = 1, fft_dim
        !NFFT likes even grids
        !The underlying FFT grids are optimized inside the nfft_init routine
        if (int(nn(ii)/2)*2 /= nn(ii) .and. (fft_optimize .and. optimize(ii)))&
          nn(ii)=nn(ii)+1
      end do
 
    case (fftlib_pnfft)
 
      do ii = 1, fft_dim
        !also PNFFT likes even grids
        if (int(nn(ii)/2)*2 /= nn(ii)) nn(ii) = nn(ii) + 1
      end do
 
      if (fft_dim < 3) then
        call messages_not_implemented('PNFFT support for dimension < 3')
      end if
 
    case default
 
      if (fft_dim < 3 .and. library_ == fftlib_pfft) then
        call messages_not_implemented('PFFT support for dimension < 3')
      end if
 
      ! FFT optimization
      if (any(optimize_parity(1:fft_dim) > 1)) then
        message(1) = "Internal error in fft_init: optimize_parity must be negative, 0, or 1."
        call messages_fatal(1)
      end if
 
      do ii = 1, fft_dim
        call loct_fft_optimize(nn_temp(ii), optimize_parity(ii))
        if (fft_optimize .and. optimize(ii)) nn(ii) = nn_temp(ii)
      end do
 
    end select
 
    ! find out if fft has already been allocated
    jj = 0
    do ii = fft_max, 1, -1
      if (fft_refs(ii) /= fft_null) then
        if (all(nn(1:dim) == fft_array(ii)%rs_n_global(1:dim)) .and. type == fft_array(ii)%type &
          .and. library_ == fft_array(ii)%library .and. library_ /= fftlib_nfft &
          .and. library_ /= fftlib_pnfft &
          .and. this%aligned_memory .eqv. fft_array(ii)%aligned_memory) then
 
          ! NFFT and PNFFT plans are always allocated from scratch since they
          ! are very likely to be different
          this = fft_array(ii)              ! return a copy
          fft_refs(ii) = fft_refs(ii) + 1  ! increment the ref count
          if (present(comm)) comm = fft_array(ii)%comm ! also return the MPI communicator
          pop_sub(fft_init)
          return
        end if
      else
        jj = ii
      end if
    end do
 
    if (jj == 0) then
      message(1) = "Not enough slots for FFTs."
      message(2) = "Please increase FFT_MAX in fft.F90 and recompile."
      call messages_fatal(2)
    end if
 
    ! jj now contains an empty slot
    fft_refs(jj) = 1
    fft_array(jj)%slot     = jj
    fft_array(jj)%type     = type
    fft_array(jj)%library  = library_
    fft_array(jj)%rs_n_global(1:dim) = nn(1:dim)
    fft_array(jj)%rs_n_global(dim+1:) = 1
    nullify(fft_array(jj)%drs_data)
    nullify(fft_array(jj)%zrs_data)
    nullify(fft_array(jj)%fs_data)
 
    fft_array(jj)%aligned_memory = this%aligned_memory
 
    ! Initialize parallel communicator
    select case (library_)
    case (fftlib_pfft)
#ifdef HAVE_PFFT
      call pfft_init()
 
      call pfft_decompose(mpi_grp_%size, column_size, row_size)
 
      ierror = pfft_create_procmesh_2d(mpi_grp_%comm%MPI_VAL, column_size, row_size, fft_array(jj)%comm%MPI_VAL)
 
      if (ierror /= 0) then
        message(1) = "The number of rows and columns in PFFT processor grid is not equal to "
        message(2) = "the number of processor in the MPI communicator."
        message(3) = "Please check it."
        call messages_fatal(3)
      end if
#endif
 
    case (fftlib_pnfft)
#ifdef HAVE_PNFFT
      call pnfft_init_procmesh(fft_array(jj)%pnfft, mpi_grp_, fft_array(jj)%comm)
#endif
    case default
      fft_array(jj)%comm = mpi_comm_undefined
 
    end select
 
    if (present(comm)) comm = fft_array(jj)%comm
 
    ! Get dimentions of arrays
    select case (library_)
    case (fftlib_fftw)
      call fftw_get_dims(fft_array(jj)%rs_n_global, type == fft_real, fft_array(jj)%fs_n_global)
      fft_array(jj)%rs_n = fft_array(jj)%rs_n_global
      fft_array(jj)%fs_n = fft_array(jj)%fs_n_global
      fft_array(jj)%rs_istart = 1
      fft_array(jj)%fs_istart = 1
 
      if (this%aligned_memory) then
        call fftw_alloc_memory(fft_array(jj)%rs_n_global, type == fft_real, fft_array(jj)%fs_n_global, &
          fft_array(jj)%drs_data, fft_array(jj)%zrs_data, fft_array(jj)%fs_data)
      end if
 
    case (fftlib_pfft)
#ifdef HAVE_PFFT
      call pfft_get_dims(fft_array(jj)%rs_n_global, comm%MPI_VAL, type == fft_real, &
        alloc_size, fft_array(jj)%fs_n_global, fft_array(jj)%rs_n, &
        fft_array(jj)%fs_n, fft_array(jj)%rs_istart, fft_array(jj)%fs_istart)
      !write(*,"(6(A,3I4,/),A,I10,/)") "PFFT: rs_n_global = ",fft_array(jj)%rs_n_global,&
      !  "fs_n_global = ",fft_array(jj)%fs_n_global,&
      !  "rs_n        = ",fft_array(jj)%rs_n,&
      !  "fs_n        = ",fft_array(jj)%fs_n,&
      !  "rs_istart   = ",fft_array(jj)%rs_istart,&
      !  "fs_istart   = ",fft_array(jj)%fs_istart,&
      !  "alloc_size  = ",alloc_size
#endif
 
      ! Allocate memory. Note that PFFT may need extra memory space
      ! and that in fourier space the function will be transposed
      if (type == fft_real) then
        n_1 = max(1, fft_array(jj)%rs_n(1))
        n_2 = max(1, fft_array(jj)%rs_n(2))
        n_3 = max(1, fft_array(jj)%rs_n(3))
 
        n3 = ceiling(real(2*alloc_size)/real(n_1*n_2))
        safe_allocate(fft_array(jj)%drs_data(1:n_1, 1:n_2, 1:n3))
      else
        n3 = ceiling(real(alloc_size)/real(fft_array(jj)%rs_n(1)*fft_array(jj)%rs_n(2)))
        safe_allocate(fft_array(jj)%zrs_data(1:fft_array(jj)%rs_n(1), 1:fft_array(jj)%rs_n(2), 1:n3))
      end if
 
      n_1 = max(1, fft_array(jj)%fs_n(1))
      n_2 = max(1, fft_array(jj)%fs_n(2))
      n_3 = max(1, fft_array(jj)%fs_n(3))
 
      n3 = ceiling(real(alloc_size)/real(n_3*n_1))
      safe_allocate(fft_array(jj)%fs_data(1:n_3, 1:n_1, 1:n3))
 
    case (fftlib_accel)
      call fftw_get_dims(fft_array(jj)%rs_n_global, (type == fft_real), fft_array(jj)%fs_n_global)
      fft_array(jj)%rs_n = fft_array(jj)%rs_n_global
      fft_array(jj)%fs_n = fft_array(jj)%fs_n_global
      fft_array(jj)%rs_istart = 1
      fft_array(jj)%fs_istart = 1
 
    case (fftlib_nfft)
      fft_array(jj)%fs_n_global = fft_array(jj)%rs_n_global
      fft_array(jj)%rs_n = fft_array(jj)%rs_n_global
      fft_array(jj)%fs_n = fft_array(jj)%fs_n_global
      fft_array(jj)%rs_istart = 1
      fft_array(jj)%fs_istart = 1
 
    case (fftlib_pnfft)
      fft_array(jj)%fs_n_global = fft_array(jj)%rs_n_global
      fft_array(jj)%rs_n = fft_array(jj)%rs_n_global
      fft_array(jj)%fs_n = fft_array(jj)%fs_n_global
      fft_array(jj)%rs_istart = 1
      fft_array(jj)%fs_istart = 1
      ! indices partition is performed together with the plan preparation
 
 
    end select
 
    ! Prepare plans
    select case (library_)
    case (fftlib_fftw)
      if (.not. this%aligned_memory) then
        call fftw_prepare_plan(fft_array(jj)%planf, fft_dim, fft_array(jj)%rs_n_global, &
          type == fft_real, fftw_forward, fft_prepare_plan+fftw_unaligned)
        call fftw_prepare_plan(fft_array(jj)%planb, fft_dim, fft_array(jj)%rs_n_global, &
          type == fft_real, fftw_backward, fft_prepare_plan+fftw_unaligned)
      else
        if (type == fft_real) then
          call fftw_prepare_plan(fft_array(jj)%planf, fft_dim, fft_array(jj)%rs_n_global, &
            type == fft_real, fftw_forward, fft_prepare_plan, &
            din_=fft_array(jj)%drs_data, cout_=fft_array(jj)%fs_data)
          call fftw_prepare_plan(fft_array(jj)%planb, fft_dim, fft_array(jj)%rs_n_global, &
            type == fft_real, fftw_backward, fft_prepare_plan, &
            din_=fft_array(jj)%drs_data, cout_=fft_array(jj)%fs_data)
        else
          call fftw_prepare_plan(fft_array(jj)%planf, fft_dim, fft_array(jj)%rs_n_global, &
            type == fft_real, fftw_forward, fft_prepare_plan, &
            cin_=fft_array(jj)%zrs_data, cout_=fft_array(jj)%fs_data)
          call fftw_prepare_plan(fft_array(jj)%planb, fft_dim, fft_array(jj)%rs_n_global, &
            type == fft_real, fftw_backward, fft_prepare_plan, &
            cin_=fft_array(jj)%zrs_data, cout_=fft_array(jj)%fs_data)
        end if
      end if
 
    case (fftlib_nfft)
#ifdef HAVE_NFFT
      call nfft_copy_info(this%nfft,fft_array(jj)%nfft) !copy default parameters set in the calling routine
      call nfft_init(fft_array(jj)%nfft, nfft_options, fft_array(jj)%rs_n_global, &
        fft_dim, fft_array(jj)%rs_n_global, optimize = .true.)
#endif
    case (fftlib_pfft)
#ifdef HAVE_PFFT
      if (type == fft_real) then
        call pfft_prepare_plan_r2c(fft_array(jj)%planf, fft_array(jj)%rs_n_global, fft_array(jj)%drs_data, &
          fft_array(jj)%fs_data, fftw_forward, fft_prepare_plan, comm%MPI_VAL)
        call pfft_prepare_plan_c2r(fft_array(jj)%planb, fft_array(jj)%rs_n_global, fft_array(jj)%fs_data, &
          fft_array(jj)%drs_data, fftw_backward, fft_prepare_plan, comm%MPI_VAL)
      else
        call pfft_prepare_plan_c2c(fft_array(jj)%planf, fft_array(jj)%rs_n_global, fft_array(jj)%zrs_data, &
          fft_array(jj)%fs_data, fftw_forward, fft_prepare_plan, comm%MPI_VAL)
        call pfft_prepare_plan_c2c(fft_array(jj)%planb, fft_array(jj)%rs_n_global, fft_array(jj)%fs_data, &
          fft_array(jj)%zrs_data, fftw_backward, fft_prepare_plan, comm%MPI_VAL)
      end if
#endif
    case (fftlib_pnfft)
#ifdef HAVE_PNFFT
      call pnfft_copy_params(this%pnfft, fft_array(jj)%pnfft) ! pass default parameters like in NFFT
 
      ! NOTE:
      ! PNFFT (likewise NFFT) breaks the symmetry between real space and Fourier space
      ! by allowing the possibility to have an unstructured grid in rs and by
      ! using different parallelizations (the rs is transposed w.r.t. fs).
      ! Octopus, in fourier_space_m, uses the convention for which the mapping
      ! between rs and fs is done with a forward transform (and fs->rs with backward).
      ! This is exactly the opposite of the definitions used by all the libraries
      ! performing FFTs (PNFFT and NFFT included) [see e.g. M. Frigo, and S. G. Johnson, Proc.
      ! IEEE 93, 216-231 (2005)].
      ! While this leads to no problem on ordinary ffts where fs and rs can be exchanged
      ! it does makes a fundamental difference for PNFFT (for some reason I don`t know NFFT
      ! is still symmetric).
      ! Therefore, in order to perform rs->fs tranforms with PNFFT one should use the
      ! backward transform.
 
      call pnfft_init_plan(fft_array(jj)%pnfft, pnfft_options, comm, fft_array(jj)%fs_n_global, &
        fft_array(jj)%fs_n, fft_array(jj)%fs_istart, fft_array(jj)%rs_n, fft_array(jj)%rs_istart)
#endif
    case (fftlib_accel)
 
      fft_array(jj)%stride_rs(1) = 1
      fft_array(jj)%stride_fs(1) = 1
      do ii = 2, fft_dim
        fft_array(jj)%stride_rs(ii) = fft_array(jj)%stride_rs(ii - 1)*fft_array(jj)%rs_n(ii - 1)
        fft_array(jj)%stride_fs(ii) = fft_array(jj)%stride_fs(ii - 1)*fft_array(jj)%fs_n(ii - 1)
      end do
 
#ifdef HAVE_CUDA
      if (type == fft_real) then
        call cuda_fft_plan3d(fft_array(jj)%cuda_plan_fw, &
          fft_array(jj)%rs_n_global(3), fft_array(jj)%rs_n_global(2), fft_array(jj)%rs_n_global(1), cufft_d2z, &
          accel%cuda_stream)
        call cuda_fft_plan3d(fft_array(jj)%cuda_plan_bw, &
          fft_array(jj)%rs_n_global(3), fft_array(jj)%rs_n_global(2), fft_array(jj)%rs_n_global(1), cufft_z2d, &
          accel%cuda_stream)
      else
        call cuda_fft_plan3d(fft_array(jj)%cuda_plan_fw, &
          fft_array(jj)%rs_n_global(3), fft_array(jj)%rs_n_global(2), fft_array(jj)%rs_n_global(1), cufft_z2z, &
          accel%cuda_stream)
        call cuda_fft_plan3d(fft_array(jj)%cuda_plan_bw, &
          fft_array(jj)%rs_n_global(3), fft_array(jj)%rs_n_global(2), fft_array(jj)%rs_n_global(1), cufft_z2z, &
          accel%cuda_stream)
      end if
#endif
 
#ifdef HAVE_CLFFT
 
      ! create the plans
      call clfftcreatedefaultplan(fft_array(jj)%cl_plan_fw, accel%context%cl_context, &
        fft_dim, int(fft_array(jj)%rs_n_global, int64), status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftCreateDefaultPlan')
 
      call clfftcreatedefaultplan(fft_array(jj)%cl_plan_bw, accel%context%cl_context, &
        fft_dim, int(fft_array(jj)%rs_n_global, int64), status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftCreateDefaultPlan')
 
      ! set precision
 
      call clfftsetplanprecision(fft_array(jj)%cl_plan_fw, clfft_double, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanPrecision')
 
      call clfftsetplanprecision(fft_array(jj)%cl_plan_bw, clfft_double, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanPrecision')
 
      ! set number of transforms to 1
 
      call clfftsetplanbatchsize(fft_array(jj)%cl_plan_fw, 1_real64, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanBatchSize')
 
      call clfftsetplanbatchsize(fft_array(jj)%cl_plan_bw, 1_real64, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanBatchSize')
 
      ! set the type precision to double
 
      call clfftsetplanprecision(fft_array(jj)%cl_plan_fw, clfft_double, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanPrecision')
 
      call clfftsetplanprecision(fft_array(jj)%cl_plan_bw, clfft_double, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanPrecision')
 
 
      ! set the layout
 
      if (type == fft_real) then
 
        call clfftsetlayout(fft_array(jj)%cl_plan_fw, clfft_real, clfft_hermitian_interleaved, status)
        if (status /= clfft_success) call clfft_print_error(status, 'clfftSetLayout')
 
        call clfftsetlayout(fft_array(jj)%cl_plan_bw, clfft_hermitian_interleaved, clfft_real, status)
        if (status /= clfft_success) call clfft_print_error(status, 'clfftSetLayout')
 
      else
 
        call clfftsetlayout(fft_array(jj)%cl_plan_fw, clfft_complex_interleaved, clfft_complex_interleaved, status)
        if (status /= clfft_success) call clfft_print_error(status, 'clfftSetLayout')
 
        call clfftsetlayout(fft_array(jj)%cl_plan_bw, clfft_complex_interleaved, clfft_complex_interleaved, status)
        if (status /= clfft_success) call clfft_print_error(status, 'clfftSetLayout')
 
      end if
 
      ! set the plans as at out of place
 
      call clfftsetresultlocation(fft_array(jj)%cl_plan_fw, clfft_outofplace, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetResultLocation')
 
      call clfftsetresultlocation(fft_array(jj)%cl_plan_bw, clfft_outofplace, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetResultLocation')
 
      ! the strides
 
      call clfftsetplaninstride(fft_array(jj)%cl_plan_fw, fft_dim, int(fft_array(jj)%stride_rs, int64), status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanInStride')
 
      call clfftsetplanoutstride(fft_array(jj)%cl_plan_fw, fft_dim, int(fft_array(jj)%stride_fs, int64), status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanOutStride')
 
      call clfftsetplaninstride(fft_array(jj)%cl_plan_bw, fft_dim, int(fft_array(jj)%stride_fs, int64), status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanInStride')
 
      call clfftsetplanoutstride(fft_array(jj)%cl_plan_bw, fft_dim, int(fft_array(jj)%stride_rs, int64), status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanOutStride')
 
      ! set the scaling factors
 
      scale = 1.0_real64/(product(real(fft_array(jj)%rs_n_global(1:fft_dim), real64)))
 
      call clfftsetplanscale(fft_array(jj)%cl_plan_fw, clfft_forward, 1.0_real64, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanScale')
 
      call clfftsetplanscale(fft_array(jj)%cl_plan_fw, clfft_backward, scale, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanScale')
 
      if (type == fft_real) then
 
        call clfftsetplanscale(fft_array(jj)%cl_plan_bw, clfft_forward, 1.0_real64, status)
        if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanScale')
 
        call clfftsetplanscale(fft_array(jj)%cl_plan_bw, clfft_backward, scale, status)
        if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanScale')
 
      else
 
        call clfftsetplanscale(fft_array(jj)%cl_plan_bw, clfft_forward, scale, status)
        if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanScale')
 
        call clfftsetplanscale(fft_array(jj)%cl_plan_bw, clfft_backward, 1.0_real64, status)
        if (status /= clfft_success) call clfft_print_error(status, 'clfftSetPlanScale')
 
      end if
 
      ! now 'bake' the plans, this signals that the plans are ready to use
 
      call clfftbakeplan(fft_array(jj)%cl_plan_fw, accel%command_queue, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftBakePlan')
 
      call clfftbakeplan(fft_array(jj)%cl_plan_bw, accel%command_queue, status)
      if (status /= clfft_success) call clfft_print_error(status, 'clfftBakePlan')
 
#endif
 
    case default
      call messages_write('Invalid FFT library.')
      call messages_fatal()
    end select
 
    this = fft_array(jj)
 
    ! Write information
    if (.not. (library_ == fftlib_nfft .or. library_ == fftlib_pnfft)) then
      call messages_write('Info: FFT grid dimensions       =')
      number_points = 1
      do idir = 1, dim
        call messages_write(fft_array(jj)%rs_n_global(idir))
        if (idir < dim) call messages_write(" x ")
        ! do the multiplication in a integer(int64) to avoid overflow for large grids
        number_points = number_points * fft_array(jj)%rs_n_global(idir)
      end do
      call messages_new_line()
 
      call messages_write('      Total grid size           =')
      call messages_write(number_points)
      call messages_write(' (')
      call messages_write(number_points*8.0_real64, units = unit_megabytes, fmt = '(f9.1)')
      call messages_write(' )')
      if (any(nn(1:fft_dim) /= nn_temp(1:fft_dim))) then
        call messages_new_line()
        call messages_write('      Inefficient FFT grid. A better grid would be: ')
        do idir = 1, fft_dim
          call messages_write(nn_temp(idir))
        end do
      end if
      call messages_info()
    end if
 
    select case (library_)
    case (fftlib_pfft)
      write(message(1),'(a)') "Info: FFT library = PFFT"
      write(message(2),'(a)') "Info: PFFT processor grid"
      write(message(3),'(a, i9)') " No. of processors                = ", mpi_grp_%size
      write(message(4),'(a, i9)') " No. of columns in the proc. grid = ", column_size
      write(message(5),'(a, i9)') " No. of rows    in the proc. grid = ", row_size
      write(message(6),'(a, i9)') " The size of integer is = ", c_intptr_t
      call messages_info(6)
 
    case (fftlib_pnfft)
#ifdef HAVE_PNFFT
      call messages_write("Info: FFT library = PNFFT")
      call messages_info()
      call pnfft_write_info(fft_array(jj)%pnfft)
#endif
    case (fftlib_nfft)
#ifdef HAVE_NFFT
      call messages_write("Info: FFT library = NFFT")
      call messages_info()
      call nfft_write_info(fft_array(jj)%nfft)
#endif
    end select
 
    pop_sub(fft_init)
  end subroutine fft_init
 
  ! ---------------------------------------------------------
  subroutine fft_init_stage1(this, namespace, XX, nn)
    type(fft_t),       intent(inout) :: this
    type(namespace_t), intent(in)    :: namespace
    real(real64),      intent(in)    :: xx(:,:)
    integer, optional, intent(in)    :: nn(:)
 
    integer :: slot
 
    push_sub(fft_init_stage1)
 
    assert(size(xx,2) == 3)
 
    slot = this%slot
    select case (fft_array(slot)%library)
    case (fftlib_fftw)
      !Do nothing
    case (fftlib_nfft)
#ifdef HAVE_NFFT
      assert(present(nn))
      call nfft_precompute(fft_array(slot)%nfft, &
        xx(1:nn(1),1), xx(1:nn(2),2), xx(1:nn(3),3))
#endif
    case (fftlib_pfft)
      !Do nothing
    case (fftlib_accel)
      !Do nothing
    case (fftlib_pnfft)
#ifdef HAVE_PNFFT
      call pnfft_set_sp_nodes(fft_array(slot)%pnfft, namespace, xx)
#endif
    case default
      call messages_write('Invalid FFT library.')
      call messages_fatal()
    end select
 
 
 
    pop_sub(fft_init_stage1)
  end subroutine fft_init_stage1
  ! ---------------------------------------------------------
  subroutine fft_end(this)
    type(fft_t), intent(inout) :: this
 
    integer :: ii
#ifdef HAVE_CLFFT
    integer :: status
#endif
 
    push_sub(fft_end)
 
    ii = this%slot
    if (fft_refs(ii) == fft_null) then
      message(1) = "Trying to deallocate FFT that has not been allocated."
      call messages_warning(1)
    else
      if (fft_refs(ii) > 1) then
        fft_refs(ii) = fft_refs(ii) - 1
      else
        select case (fft_array(ii)%library)
        case (fftlib_fftw)
          call fftw_destroy_plan(fft_array(ii)%planf)
          call fftw_destroy_plan(fft_array(ii)%planb)
 
          if (this%aligned_memory) then
            call fftw_free_memory(this%type == fft_real, &
              fft_array(ii)%drs_data, fft_array(ii)%zrs_data, fft_array(ii)%fs_data)
          end if
 
        case (fftlib_pfft)
#ifdef HAVE_PFFT
          call pfft_destroy_plan(fft_array(ii)%planf)
          call pfft_destroy_plan(fft_array(ii)%planb)
#endif
          safe_deallocate_p(fft_array(ii)%drs_data)
          safe_deallocate_p(fft_array(ii)%zrs_data)
          safe_deallocate_p(fft_array(ii)%fs_data)
 
        case (fftlib_accel)
#ifdef HAVE_CUDA
          call cuda_fft_destroy(fft_array(ii)%cuda_plan_fw)
          call cuda_fft_destroy(fft_array(ii)%cuda_plan_bw)
#endif
#ifdef HAVE_CLFFT
          call clfftdestroyplan(fft_array(ii)%cl_plan_fw, status)
          call clfftdestroyplan(fft_array(ii)%cl_plan_bw, status)
#endif
 
        case (fftlib_nfft)
#ifdef HAVE_NFFT
          call nfft_end(fft_array(ii)%nfft)
#endif
        case (fftlib_pnfft)
#ifdef HAVE_PNFFT
          call pnfft_end(fft_array(ii)%pnfft)
#endif
        end select
        fft_refs(ii) = fft_null
      end if
    end if
    this%slot = 0
 
    pop_sub(fft_end)
  end subroutine fft_end
 
  ! ---------------------------------------------------------
  subroutine fft_copy(fft_i, fft_o)
    type(fft_t), intent(in)    :: fft_i
    type(fft_t), intent(inout) :: fft_o
 
    push_sub(fft_copy)
 
    if (fft_o%slot > 0) then
      call fft_end(fft_o)
    end if
    assert(fft_i%slot >= 1.and.fft_i%slot <= fft_max)
    assert(fft_refs(fft_i%slot) > 0)
 
    fft_o = fft_i
    fft_refs(fft_i%slot) = fft_refs(fft_i%slot) + 1
 
    pop_sub(fft_copy)
  end subroutine fft_copy
 
  ! ---------------------------------------------------------
  subroutine fft_get_dims(fft, rs_n_global, fs_n_global, rs_n, fs_n, rs_istart, fs_istart)
    type(fft_t), intent(in)  :: fft
    integer,     intent(out) :: rs_n_global(1:3)
    integer,     intent(out) :: fs_n_global(1:3)
    integer,     intent(out) :: rs_n(1:3)
    integer,     intent(out) :: fs_n(1:3)
    integer,     intent(out) :: rs_istart(1:3)
    integer,     intent(out) :: fs_istart(1:3)
 
    integer :: slot
 
    push_sub(fft_get_dims)
 
    slot = fft%slot
    rs_n_global(1:3) = fft_array(slot)%rs_n_global(1:3)
    fs_n_global(1:3) = fft_array(slot)%fs_n_global(1:3)
    rs_n(1:3) = fft_array(slot)%rs_n(1:3)
    fs_n(1:3) = fft_array(slot)%fs_n(1:3)
    rs_istart(1:3) = fft_array(slot)%rs_istart(1:3)
    fs_istart(1:3) = fft_array(slot)%fs_istart(1:3)
 
    pop_sub(fft_get_dims)
  end subroutine fft_get_dims
 
  ! ---------------------------------------------------------
  pure function pad_feq(ii, nn, mode)
    integer, intent(in) :: ii,nn
    logical, intent(in) :: mode
    integer :: pad_feq
 
    ! no push_sub: called too frequently
 
    if (mode) then      ! index to frequency number
      if (ii <= nn/2 + 1) then
        pad_feq = ii - 1
      else
        pad_feq = ii - nn - 1
      end if
    else
      if (ii >= 0) then
        pad_feq = ii + 1
      else
        pad_feq = ii + nn + 1
      end if
    end if
 
  end function pad_feq
 
  ! -------------------------------------------------------
 
  integer function fft_size(size, factors, parity)
    integer, intent(in) :: size
    integer, intent(in) :: factors(:)
    integer, intent(in) :: parity
 
    integer :: nfactors
    integer :: nondiv
    integer, allocatable :: exponents(:)
 
    push_sub(fft_size)
 
    nfactors = ubound(factors, dim = 1)
 
    safe_allocate(exponents(1:nfactors))
 
    fft_size = size
    do
      call get_exponents(fft_size, nfactors, factors, exponents, nondiv)
      if (nondiv == 1 .and. mod(fft_size, 2) == parity) exit
      fft_size = fft_size + 1
    end do
 
    safe_deallocate_a(exponents)
 
    pop_sub(fft_size)
  end function fft_size
 
  ! -------------------------------------------------------
 
  subroutine get_exponents(num, nfactors, factors, exponents, nondiv)
    integer, intent(in)  :: num
    integer, intent(in)  :: nfactors
    integer, intent(in)  :: factors(:)
    integer, intent(out) :: exponents(:)
    integer, intent(out) :: nondiv
 
    integer :: ifactor
 
    push_sub(get_exponents)
 
    nondiv = num
    do ifactor = 1, nfactors
      exponents(ifactor) = 0
      do
        if (mod(nondiv, factors(ifactor)) /= 0) exit
        nondiv = nondiv/factors(ifactor)
        exponents(ifactor) = exponents(ifactor) + 1
      end do
    end do
 
    pop_sub(get_exponents)
  end subroutine get_exponents
 
 
  ! ----------------------------------------------------------
 
  subroutine fft_operation_count(fft)
    type(fft_t), intent(in)  :: fft
 
    real(real64) :: fullsize
 
    push_sub(fft_operation_count)
 
    fullsize = product(real(fft%fs_n(1:3), real64))
    call profiling_count_operations(5.0_real64*fullsize*log(fullsize)/log(m_two))
 
    pop_sub(fft_operation_count)
  end subroutine fft_operation_count
 
  !-----------------------------------------------------------------
  subroutine fft_gg_transform(gg_in, temp, periodic_dim, latt, qq, gg, modg2)
    integer,                 intent(in)    :: gg_in(:)
    real(real64),            intent(in)    :: temp(:)
    integer,                 intent(in)    :: periodic_dim
    type(lattice_vectors_t), intent(in)    :: latt
    real(real64),            intent(in)    :: qq(:)
    real(real64),            intent(inout) :: gg(:)
    real(real64),            intent(out)   :: modg2
 
    ! no PUSH_SUB, called too frequently
 
    gg(1:3) = gg_in(1:3)
    gg(1:periodic_dim) = gg(1:periodic_dim) + qq(1:periodic_dim)
    gg(1:3) = gg(1:3) * temp(1:3)
    gg(1:3) = matmul(latt%klattice_primitive(1:3,1:3),gg(1:3))
    modg2 = sum(gg(1:3)**2)
 
  end subroutine fft_gg_transform
 
  ! ----------------------------------------------------------
 
  real(real64) pure function fft_scaling_factor(fft) result(scaling_factor)
    type(fft_t), intent(in)  :: fft
 
    ! for the moment this factor is handled by the backwards transform for most libraries
    scaling_factor = m_one
 
    select case (fft_array(fft%slot)%library)
    case (fftlib_accel)
#ifdef HAVE_CUDA
      scaling_factor = m_one/real(fft_array(fft%slot)%rs_n_global(1), real64)
      scaling_factor = scaling_factor/real(fft_array(fft%slot)%rs_n_global(2), real64)
      scaling_factor = scaling_factor/real(fft_array(fft%slot)%rs_n_global(3), real64)
#endif
    end select
 
  end function fft_scaling_factor
 
  ! ----------------------------------------------------------
  !
  ! Inspired by the routine bounds from Abinit
  real(real64) function fft_get_ecut_from_box(box_dim, fs_istart, latt, gspacing, periodic_dim, qq) result(ecut)
    integer,                 intent(in) :: box_dim(:)
    integer,                 intent(in) :: fs_istart(:)
    type(lattice_vectors_t), intent(in) :: latt
    real(real64),            intent(in) :: gspacing(:)
    integer,                 intent(in) :: periodic_dim
    real(real64),            intent(in) :: qq(:)
 
    integer :: lx, ix, iy, iz, idir, idir2, idir3
    real(real64) :: dminsq, gg(3), modg2
    integer :: box_dim_(3), ixx(3)
    integer :: ming(3), maxg(3)
 
    ! no PUSH_SUB, called too frequently
 
    assert(periodic_dim > 0)
 
    box_dim_(1:periodic_dim) = box_dim(1:periodic_dim)
    if (periodic_dim < 3) box_dim_(periodic_dim+1:3) = 1
 
    ! We first need to remove asymetric planes for the case of even FFT grids
    ming = 1
    maxg = 1
    do idir = 1, periodic_dim
      do lx = 1, box_dim(idir)
        ix = fs_istart(idir) + lx - 1
        ixx(idir) = pad_feq(ix, box_dim(idir), .true.)
        ming(idir) = min(ming(idir), ixx(idir))
        maxg(idir) = max(maxg(idir), ixx(idir))
      end do
      maxg(idir) = min(abs(ming(idir)), maxg(idir))
    end do
 
    ! Given the boundaries, we can search the min distance, which gives us the the cutoff energy
    dminsq = m_huge
    do idir = 1, periodic_dim
      idir2 = mod(idir, 3)+1
      idir3 = mod(idir+1, 3)+1
 
      ! Negative plane
      ixx(idir) = -maxg(idir)
      do iy = -maxg(idir2), maxg(idir2)
        ixx(idir2) = iy
        do iz = -maxg(idir3), maxg(idir3)
          ixx(idir3) = iz
          call fft_gg_transform(ixx, gspacing, periodic_dim, latt, qq, gg, modg2)
          dminsq = min(dminsq, sum(gg(1:periodic_dim)**2))
        end do
      end do
      ! Positive plane
      ixx(idir) = maxg(idir)
      do iy = -maxg(idir2), maxg(idir2)
        ixx(idir2) = iy
        do iz = -maxg(idir3), maxg(idir3)
          ixx(idir3) = iz
          call fft_gg_transform(ixx, gspacing, periodic_dim, latt, qq, gg, modg2)
          dminsq = min(dminsq, sum(gg(1:periodic_dim)**2))
        end do
      end do
    end do
 
    ecut = m_half * dminsq
 
  end function fft_get_ecut_from_box
 
#include "undef.F90"
#include "real.F90"
#include "fft_inc.F90"
 
#include "undef.F90"
#include "complex.F90"
#include "fft_inc.F90"
 
end module fft_oct_m
 
!! Local Variables:
!! mode: f90
!! coding: utf-8
!! End: