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Exponentials.F90
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Exponentials.F90
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! MIT License
!
! Copyright (c) 2018-2021 Florian Goth
!
! Permission is hereby granted, free of charge, to any person obtaining a copy
! of this software and associated documentation files (the "Software"), to deal
! in the Software without restriction, including without limitation the rights
! to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
! copies of the Software, and to permit persons to whom the Software is
! furnished to do so, subject to the following conditions:
!
! The above copyright notice and this permission notice shall be included in
! all copies or substantial portions of the Software.
!
! THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
! OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
! FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
! THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
! LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
! FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
! DEALINGS IN THE SOFTWARE.
module Exponentials_mod
Use ZeroDiagSingleColExp_mod
Use HomogeneousSingleColExp_mod
Use TraceLessSingleColExp_mod
Use GeneralSingleColExp_mod
implicit none
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> A wrapper type such that Fortran can hold an array of
!> base class pointers.
!--------------------------------------------------------------------
type :: SingleColeExpBaseWrapper
class(SingleColExpBase), pointer :: dat => null()
end type
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> This holds together a set of exponentials that, if applied in the
!> correct order, approximate e^A to first order.
!> It provides functions for matrix-matrix and matrix-vector
!> multiplications in transposed and non-transposed manner.
!--------------------------------------------------------------------
type :: EulerExp
integer :: nrofcols ! The number of colors
Type(SingleColeExpBaseWrapper), allocatable :: singleexps(:) ! an array of pointers to the actual polymorphic classes that do the work
contains
procedure :: init => EulerExp_init
procedure :: dealloc => EulerExp_dealloc
procedure :: vecmult => EulerExp_vecmult
procedure :: vecmult_T => EulerExp_vecmult_T
procedure :: lmult => EulerExp_lmult
procedure :: lmultinv => EulerExp_lmultinv
procedure :: rmult => EulerExp_rmult
procedure :: rmultinv => EulerExp_rmultinv
procedure :: rmult_T => EulerExp_rmult_T
procedure :: lmult_T => EulerExp_lmult_T
procedure :: adjointaction => EulerExp_adjointaction
procedure :: adjoint_over_two => EulerExp_adjoint_over_two
procedure :: adjoint_over_two_T => EulerExp_adjoint_over_two_T
procedure :: rmultinv_T => EulerExp_rmultinv_T
procedure :: lmultinv_T => EulerExp_lmultinv_T
end type EulerExp
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> This holds together a set of Euler exponentials
!> and applies them in the correct order to obtain higher order
!> approximations.
!> Note that due to the use of higher-order approximations we have successive
!> Strang pairings. Hence the Euler type method needs to be treated separately.
!--------------------------------------------------------------------
type :: FullExp
integer :: method
integer :: evals
type(EulerExp), allocatable :: stages(:)
contains
procedure :: init => FullExp_init
procedure :: dealloc => FullExp_dealloc
procedure :: vecmult => FullExp_vecmult
procedure :: vecmult_T => FullExp_vecmult_T
procedure :: lmult => FullExp_lmult
procedure :: lmultinv => FullExp_lmultinv
procedure :: rmult => FullExp_rmult
procedure :: rmultinv => FullExp_rmultinv
procedure :: lmult_T => FullExp_lmult_T
procedure :: adjoint_over_two => FullExp_adjoint_over_two
end type FullExp
contains
subroutine FullExp_init(this, nodes, usedcolors, dcols, method, weight)
class(FullExp) :: this
type(node), dimension(:), intent(in) :: nodes
integer, intent(in) :: usedcolors, method
real(kind=kind(0.D0)), intent(in), allocatable, dimension(:, :) :: dcols
real (kind=kind(0.d0)), intent(in) :: weight
real (kind=kind(0.d0)) :: tmp
this%method = method
#ifndef NDEBUG
write(*,*) "Setting up Full Checkerboard exponential."
#endif
select case (method)
case (2)! Strang
this%evals = 2
allocate(this%stages(this%evals))
tmp = 1.D0/2.D0*weight
call this%stages(1)%init(nodes, usedcolors, dcols, tmp)
call this%stages(2)%init(nodes, usedcolors, dcols, tmp)
case (3)! SE_2 2, Blanes
this%evals = 4
allocate(this%stages(this%evals))
tmp = 0.21178*weight
call this%stages(1)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.28822*weight
call this%stages(2)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.28822*weight
call this%stages(3)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.21178*weight
call this%stages(4)%init(nodes, usedcolors, dcols, tmp)
case (4)! S_3 4, Yoshida, Neri, Suzuki, 1990
this%evals = 6
allocate(this%stages(this%evals))
tmp = 0.6756035959798*weight
call this%stages(1)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.6756035959798*weight
call this%stages(2)%init(nodes, usedcolors, dcols, tmp)
tmp = -0.8512071919597*weight
call this%stages(3)%init(nodes, usedcolors, dcols, tmp)
tmp = -0.8512071919597*weight
call this%stages(4)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.6756035959798*weight
call this%stages(5)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.6756035959798*weight
call this%stages(6)%init(nodes, usedcolors, dcols, tmp)
case (5)! SE_6 4, Blanes, Blanes and Moan 2002
this%evals = 12
allocate(this%stages(this%evals))
tmp = 0.079203696431195694249716154899943D0 *weight
call this%stages(1)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.13031141018216631233261892930386D0 *weight
call this%stages(2)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.22286149586760771457161212083520D0 *weight
call this%stages(3)%init(nodes, usedcolors, dcols, tmp)
tmp = -0.36671326904742572450057735977680D0 *weight
call this%stages(4)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.32464818868970622689484883949262D0 *weight
call this%stages(5)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.1096884778767497764517813152452D0 *weight
call this%stages(6)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.1096884778767497764517813152452D0 *weight
call this%stages(7)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.3246481886897062268948488394926D0 *weight
call this%stages(8)%init(nodes, usedcolors, dcols, tmp)
tmp = -0.3667132690474257245005773597768D0 *weight
call this%stages(9)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.2228614958676077145716121208352D0 *weight
call this%stages(10)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.1303114101821663123326189293039D0 *weight
call this%stages(11)%init(nodes, usedcolors, dcols, tmp)
tmp = 0.0792036964311956942497161548999D0 *weight
call this%stages(12)%init(nodes, usedcolors, dcols, tmp)
end select
end subroutine FullExp_init
subroutine FullExp_dealloc(this)
class(FullExp) :: this
integer :: i
do i = 1, this%evals
call this%stages(i)%dealloc()
enddo
deallocate(this%stages)
end subroutine FullExp_dealloc
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> This function multiplies this full exponential with a vector
!
!> @param[in] this The exponential opbject
!> @param[in] vec The vector that we multiply
!--------------------------------------------------------------------
subroutine FullExp_vecmult(this, vec)
class(FullExp) :: this
complex(kind=kind(0.D0)), dimension(:), intent(inout) :: vec
integer :: i
do i = 1, this%evals
call this%stages(i)%vecmult(vec)
enddo
end subroutine FullExp_vecmult
subroutine FullExp_vecmult_T(this, vec)
class(FullExp) :: this
complex(kind=kind(0.D0)), dimension(:), intent(inout) :: vec
integer :: i
do i = this%evals, 1, -1
call this%stages(i)%vecmult_T(vec)
enddo
end subroutine FullExp_vecmult_T
subroutine FullExp_lmult(this, mat)
class(FullExp) :: this
complex(kind=kind(0.D0)), intent(inout), contiguous :: mat(:,:)
integer :: i
do i = this%evals-1, 1, -2
call this%stages(i+1)%lmult_T(mat)
call this%stages(i)%lmult(mat)
enddo
end subroutine FullExp_lmult
subroutine FullExp_adjoint_over_two(this, mat)
class(FullExp) :: this
complex(kind=kind(0.D0)), intent(inout) :: mat(:,:)
integer :: i
do i = this%evals-1, 1, -2
call this%stages(i+1)%adjoint_over_two_T(mat)
call this%stages(i)%adjoint_over_two(mat)
enddo
end subroutine FullExp_adjoint_over_two
subroutine FullExp_lmultinv(this, mat)
class(FullExp) :: this
complex(kind=kind(0.D0)), intent(inout), contiguous :: mat(:,:)
integer :: i
do i = 1, this%evals, 2
call this%stages(i)%lmultinv(mat)
call this%stages(i+1)%lmultinv_T(mat)
enddo
end subroutine FullExp_lmultinv
subroutine FullExp_rmult(this, mat)
class(FullExp) :: this
complex(kind=kind(0.D0)), dimension(:, :), intent(inout) :: mat
integer :: i
do i = 1, this%evals,2
call this%stages(i)%rmult(mat)
call this%stages(i+1)%rmult_T(mat)
enddo
end subroutine FullExp_rmult
subroutine FullExp_rmultinv(this, mat)
class(FullExp) :: this
complex(kind=kind(0.D0)), intent(inout) :: mat(:,:)
integer :: i
do i = this%evals-1, 1, -2
call this%stages(i+1)%rmultinv_T(mat)
call this%stages(i)%rmultinv(mat)
enddo
end subroutine FullExp_rmultinv
subroutine FullExp_lmult_T(this, mat)
class(FullExp) :: this
complex(kind=kind(0.D0)), dimension(:, :), intent(inout) :: mat
integer :: i
do i = 1, this%evals, 2
call this%stages(i)%lmult_T(mat)
call this%stages(i+1)%lmult(mat)
enddo
end subroutine FullExp_lmult_T
subroutine EulerExp_dealloc(this)
class(EulerExp) :: this
integer :: i
do i = 1, this%nrofcols
call this%singleexps(i)%dat%dealloc()
deallocate(this%singleexps(i)%dat)
enddo
deallocate(this%singleexps)
end subroutine EulerExp_dealloc
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> This function multiplies this Euler exponential with a vector.
!
!> @param[in] this The exponential opbject
!> @param[in] vec The vector that we multiply
!--------------------------------------------------------------------
subroutine EulerExp_vecmult(this, vec)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:) :: vec
integer :: i
do i = 1, this%nrofcols
call this%singleexps(i)%dat%vecmult(vec)
enddo
end subroutine EulerExp_vecmult
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> This function multiplies this transposed full exponential,
!> i.e reverses the order of application of exponentials, with a vector.
!
!> @param[in] this The exponential opbject
!> @param[in] vec The vector that we multiply
!--------------------------------------------------------------------
subroutine EulerExp_vecmult_T(this, vec)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:) :: vec
integer :: i
do i = this%nrofcols, 1, -1
call this%singleexps(i)%dat%vecmult(vec)
enddo
end subroutine EulerExp_vecmult_T
subroutine EulerExp_lmultinv(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :), contiguous :: mat
integer :: i
do i = 1, this%nrofcols
call this%singleexps(i)%dat%lmultinv(mat)
enddo
end subroutine EulerExp_lmultinv
subroutine EulerExp_lmult(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :), contiguous :: mat
integer :: i
do i = this%nrofcols, 1, -1
call this%singleexps(i)%dat%lmult(mat)
enddo
end subroutine EulerExp_lmult
subroutine EulerExp_adjointaction(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = this%nrofcols, 1, -1
call this%singleexps(i)%dat%adjointaction(mat)
enddo
end subroutine EulerExp_adjointaction
subroutine EulerExp_adjoint_over_two(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = this%nrofcols, 1, -1
call this%singleexps(i)%dat%adjoint_over_two(mat)
enddo
end subroutine EulerExp_adjoint_over_two
subroutine EulerExp_adjoint_over_two_T(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = 1, this%nrofcols
call this%singleexps(i)%dat%adjoint_over_two(mat)
enddo
end subroutine EulerExp_adjoint_over_two_T
subroutine EulerExp_rmult(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = 1, this%nrofcols
call this%singleexps(i)%dat%rmult(mat)
enddo
end subroutine EulerExp_rmult
subroutine EulerExp_rmultinv(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = this%nrofcols, 1, -1
call this%singleexps(i)%dat%rmultinv(mat)
enddo
end subroutine EulerExp_rmultinv
subroutine EulerExp_rmult_T(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = this%nrofcols, 1, -1
call this%singleexps(i)%dat%rmult(mat)
enddo
end subroutine EulerExp_rmult_T
subroutine EulerExp_rmultinv_T(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = 1, this%nrofcols
call this%singleexps(i)%dat%rmultinv(mat)
enddo
end subroutine EulerExp_rmultinv_T
subroutine EulerExp_lmult_T(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = 1, this%nrofcols
call this%singleexps(i)%dat%lmult(mat)
enddo
end subroutine EulerExp_lmult_T
subroutine EulerExp_lmultinv_T(this, mat)
class(EulerExp) :: this
complex(kind=kind(0.D0)), dimension(:, :) :: mat
integer :: i
do i = this%nrofcols, 1, -1
call this%singleexps(i)%dat%lmultinv(mat)
enddo
end subroutine EulerExp_lmultinv_T
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> A function to determine the equality of two floating point numbers.
!> @param[in] a first number
!> @param[in] b second number
!> @return true if the numbers are equal up to 10^-15 of relative deviation.
!--------------------------------------------------------------------
function fpequal(a, b) result(isequal)
real(kind=kind(0.D0)), intent(in) :: a, b
logical :: isequal
isequal = .true.
if(abs(a - b) > max(abs(a), abs(b)) * 1E-15) then
isequal = .false.
endif
end function
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> This function classifies the diagonal.
!> @param[in] mys a vector containing the chemical potentials.
!> @return 0: ZeroDiag, 1:HomogeneousSingleColExp, 2:Traceless: 3: General
!--------------------------------------------------------------------
function determinediagtype(nodes, nrnodes, mys) result(diagtype)
real(kind=kind(0.D0)), intent(in), dimension(:) :: mys
type(node), dimension(:), intent(in) :: nodes
integer, intent(in) :: nrnodes
integer :: diagtype
integer :: i
real(kind=kind(0.D0)) :: localzero
logical :: isequal, iszero, istraceless
! check for pairwise equality
isequal = .true.
do i = 1, nrnodes
if ( .not. fpequal(mys(nodes(i)%x), mys(nodes(i)%y ) )) then
isequal = .false.
endif
enddo
if(isequal) then
! check whether they are as good as zero.
iszero = .true.
do i = 1, nrnodes
localzero = 1E-15*abs(nodes(i)%axy)*sqrt(2.D0*(mys(nodes(i)%x)/abs(nodes(i)%axy) )**2 + 1.D0)
if (abs(mys(nodes(i)%x)) > localzero) iszero = .false.
enddo
if (iszero) then
diagtype = 0
else
diagtype = 1
endif
else
! check whether all blocks are traceless
istraceless = .true.
do i = 1, nrnodes
localzero = 1E-15*frobnorm(mys(nodes(i)%x), mys(nodes(i)%y), nodes(i)%axy)
if (abs(mys(nodes(i)%x) + mys(nodes(i)%y)) > localzero) istraceless = .false.
enddo
if (istraceless) then
diagtype = 2
else
diagtype = 3
endif
endif
end function
!--------------------------------------------------------------------
!> @author
!> Florian Goth
!
!> @brief
!> This function creates an exponential object from an array of nodes.
!
!> @param this The exponential object
!> @param[in] nodes The array of nodes
!> @param[in] usedcolors the number of used colors/terms in
!> the decomposition.
!> @param[in] dcols a usedcolors x ndim array that contains for each color the diagonal entries to be used.
!> @param[in] weight a prefactor of the exponent.
!--------------------------------------------------------------------
subroutine EulerExp_init(this, nodes, usedcolors, dcols, weight)
class(EulerExp) :: this
type(node), dimension(:), intent(in) :: nodes
integer, intent(in) :: usedcolors
real(kind=kind(0.D0)), intent(in), allocatable, dimension(:, :) :: dcols
real(kind=kind(0.d0)), intent(in) :: weight
integer, dimension(:), allocatable :: nredges ! An array for determining how many edges are there for each color
integer, dimension(:), allocatable :: edgectr ! A helper array for counting.
integer :: i, maxedges
! ! integer :: k
! ! character(64) :: filename
type(node), dimension(:, :), allocatable :: colsepnodes! An array of nodes separated by color
class(ZeroDiagSingleColExp), pointer :: zerodiagexp => null()
class(HomogeneousSingleColExp), pointer :: homexp => null()
class(TraceLessSingleColExp), pointer :: tracelessexp => null()
class(GeneralSingleColExp), pointer :: generalexp => null()
#ifndef NDEBUG
write(*,*) "Setting up Euler Checkerboard exponential."
#endif
! Determine the number of matrix entries in each family
allocate (nredges(usedcolors), edgectr(usedcolors))
nredges = 0
this%nrofcols = usedcolors
do i = 1, size(nodes)
nredges(nodes(i)%col) = nredges(nodes(i)%col) + 1
enddo
maxedges = maxval(nredges)
edgectr = 1
allocate(colsepnodes(usedcolors, maxedges))
do i = 1, size(nodes)
colsepnodes(nodes(i)%col, edgectr(nodes(i)%col)) = nodes(i)
edgectr(nodes(i)%col) = edgectr(nodes(i)%col) + 1
enddo
! ! Useful for generating input for mathematica
! ! do i = 1, usedcolors
! ! write (filename, "(A6,I3)") "matrix", i
! ! open(unit=5,file=filename)
! ! do k = 1, nredges(i)
! ! write (5, *) "{{", colsepnodes(i, k)%x, ",",colsepnodes(i, k)%y,"} -> ", dble(colsepnodes(i, k)%axy), "}"
! ! enddo
! ! close(unit=5)
! ! enddo
! Now that we have properly separated which entry of a matrix belongs to
! which color we can create an exponential for each color that exploits
! the structure, that the color decomposition creates strictly sparse matrices.
allocate(this%singleexps(usedcolors))
do i = 1, usedcolors
! In each color we have to determine which optimizations are possible
select case(determinediagtype( colsepnodes(i, :), nredges(i), dcols(i,:) ))
case(0)
allocate(zerodiagexp)
this%singleexps(i)%dat => zerodiagexp
case(1)
allocate(homexp)
this%singleexps(i)%dat => homexp
case(2)
allocate(tracelessexp)
this%singleexps(i)%dat => tracelessexp
case(3)
allocate(generalexp)
this%singleexps(i)%dat => generalexp
end select
call this%singleexps(i)%dat%init(colsepnodes(i, :), nredges(i), dcols(i, :), weight)
enddo
deallocate(nredges, edgectr, colsepnodes)
end subroutine EulerExp_init
end module Exponentials_mod