!------------------------------------------------------------------------
function Kronecker_delta(i,j) result(delta)

! Kronecker Delta

  implicit none

! Input variables

  integer,intent(in)            :: i,j

! Output variables

  double precision              :: delta

  if(i == j) then
    delta = 1d0
  else
    delta = 0d0
  endif

end function Kronecker_delta

function KroneckerDelta(i,j) result(delta)

! Kronecker Delta

  implicit none

! Input variables

  integer,intent(in)            :: i,j


! Output variables

  integer                       :: delta

  if(i == j) then
    delta = 1
  else
    delta = 0
  endif

end function KroneckerDelta

!------------------------------------------------------------------------
subroutine matout(m,n,A)

! Print the MxN array A

  implicit none

  integer,parameter             :: ncol = 5
  double precision,parameter    :: small = 1d-10
  integer,intent(in)            :: m,n
  double precision,intent(in)   :: A(m,n)
  double precision              :: B(ncol)
  integer                       :: ilower,iupper,num,i,j
  
  do ilower=1,n,ncol
    iupper = min(ilower + ncol - 1,n)
    num = iupper - ilower + 1
    write(*,'(3X,10(9X,I6))') (j,j=ilower,iupper)
    do i=1,m
      do j=ilower,iupper
        B(j-ilower+1) = A(i,j)
      enddo
      do j=1,num
        if(abs(B(j)) < small) B(j) = 0d0
      enddo
      write(*,'(I7,10F15.8)') i,(B(j),j=1,num)
    enddo
  enddo

end subroutine matout

!------------------------------------------------------------------------
subroutine trace_vector(n,v,Tr)

! Calculate the trace of the vector v of length n
!!! Please use the intrinsic fortran sum()  !!!

  implicit none

! Input variables

  integer,intent(in)            :: n
  double precision,intent(in)   :: v(n)

! Local variables

  integer                       :: i

! Output variables

  double precision,intent(out)  :: Tr

  Tr = 0d0
  do i=1,n
    Tr = Tr + v(i)
  enddo

end subroutine trace_vector

!------------------------------------------------------------------------
function trace_matrix(n,A) result(Tr)

! Calculate the trace of the square matrix A

  implicit none

! Input variables

  integer,intent(in)            :: n
  double precision,intent(in)   :: A(n,n)

! Local variables

  integer                       :: i

! Output variables

  double precision              :: Tr

  Tr = 0d0
  do i=1,n
    Tr = Tr + A(i,i)
  enddo

end function trace_matrix

!------------------------------------------------------------------------
subroutine compute_error(nData,Mean,Var,Error)

! Calculate the statistical error

  implicit none

! Input variables

  double precision,intent(in)   :: nData,Mean(3)

! Output variables

  double precision,intent(out)  :: Error(3)
  double precision,intent(inout):: Var(3)
  
  Error = sqrt((Var-Mean**2/nData)/nData/(nData-1d0))

end subroutine compute_error

!------------------------------------------------------------------------
subroutine identity_matrix(N,A)

! Set the matrix A to the identity matrix

  implicit none

! Input variables

  integer,intent(in)            :: N

! Local viaruabkes

  integer                       :: i

! Output variables

  double precision,intent(out)  :: A(N,N)
  
  A = 0d0

  do i=1,N
    A(i,i) = 1d0
  enddo
     
end subroutine identity_matrix

!------------------------------------------------------------------------
subroutine prepend(N,M,A,b)

! Prepend the vector b of size N into the matrix A of size NxM

  implicit none

! Input variables

  integer,intent(in)            :: N,M
  double precision,intent(in)   :: b(N)

! Local viaruabkes

  integer                       :: i,j

! Output variables

  double precision,intent(out)  :: A(N,M)


! print*,'b in append'
! call matout(N,1,b)

  do i=1,N
    do j=M-1,1,-1
      A(i,j+1) = A(i,j)
    enddo
    A(i,1) = b(i)
  enddo

end subroutine prepend

!------------------------------------------------------------------------
subroutine append(N,M,A,b)

! Append the vector b of size N into the matrix A of size NxM

  implicit none

! Input variables

  integer,intent(in)            :: N,M
  double precision,intent(in)   :: b(N)

! Local viaruabkes

  integer                       :: i,j

! Output variables

  double precision,intent(out)  :: A(N,M)

  do i=1,N
    do j=2,M
      A(i,j-1) = A(i,j)
    enddo
    A(i,M) = b(i)
  enddo

end subroutine append

!------------------------------------------------------------------------
subroutine AtDA(N,A,D,B)

! Perform B = At.D.A where A is a NxN matrix and D is a diagonal matrix given 
! as a vector of length N

  implicit none

! Input variables

  integer,intent(in)            :: N
  double precision,intent(in)   :: A(N,N),D(N)

! Local viaruabkes

  integer                       :: i,j,k

! Output variables

  double precision,intent(out)  :: B(N,N)

  B = 0d0

  do i=1,N
    do j=1,N
      do k=1,N
        B(i,k) = B(i,k) + A(j,i)*D(j)*A(j,k)
      enddo
    enddo
  enddo

end subroutine AtDA

!------------------------------------------------------------------------
subroutine ADAt(N,A,D,B)

! Perform B = A.D.At where A is a NxN matrix and D is a diagonal matrix given 
! as a vector of length N

  implicit none

! Input variables

  integer,intent(in)            :: N
  double precision,intent(in)   :: A(N,N),D(N)

! Local viaruabkes

  integer                       :: i,j,k

! Output variables

  double precision,intent(out)  :: B(N,N)

  B = 0d0

  do i=1,N
    do j=1,N
      do k=1,N
        B(i,k) = B(i,k) + A(i,j)*D(j)*A(k,j)
      enddo
    enddo
  enddo

end subroutine ADAt
!------------------------------------------------------------------------
subroutine DA(N,D,A)

! Perform A <- D.A where A is a NxN matrix and D is a diagonal matrix given 
! as a vector of length N

  implicit none

  integer,intent(in)            :: N
  integer                       :: i,j,k
  double precision,intent(in)   :: D(N)
  double precision,intent(inout):: A(N,N)

  do i=1,N
    do j=1,N
      A(i,j) = D(i)*A(i,j)
    enddo
  enddo

end subroutine DA

!------------------------------------------------------------------------
subroutine AD(N,A,D)

! Perform A <- A.D where A is a NxN matrix and D is a diagonal matrix given 
! as a vector of length N

  implicit none

  integer,intent(in)            :: N
  integer                       :: i,j,k
  double precision,intent(in)   :: D(N)
  double precision,intent(inout):: A(N,N)

  do i=1,N
    do j=1,N
      A(i,j) = A(i,j)*D(j)
    enddo
  enddo

end subroutine AD

!------------------------------------------------------------------------
subroutine print_warning(message)

! Print warning

  implicit none

  character(len=*),intent(in)            :: message

  write(*,*) message

end subroutine print_warning

!------------------------------------------------------------------------

subroutine CalcTrAB(n,A,B,Tr)

! Calculate the trace of the square matrix A.B

  implicit none

! Input variables

  integer,intent(in)            :: n
  double precision,intent(in)   :: A(n,n),B(n,n)

! Local variables

  integer                       :: i,j

! Output variables

  double precision,intent(out)  :: Tr

  Tr = 0d0
  do i=1,n
    do j=1,n
      Tr = Tr + A(i,j)*B(j,i)
    enddo
  enddo

end subroutine CalcTrAB

!------------------------------------------------------------------------

function EpsilonSwitch(i,j) result(delta)

! Epsilon function 

  implicit none

! Input variables

  integer,intent(in)            :: i,j
  integer                       :: delta

  if(i <= j) then
    delta = 1
  else
    delta = -1
  endif

end function EpsilonSwitch

!------------------------------------------------------------------------

function KappaCross(i,j,k) result(kappa)

! kappa(i,j,k) = eps(i,j) delta(i,k) - eps(k,i) delta(i,j)

  implicit none

! Input variables

  integer,intent(in)            :: i,j,k

! Local variables 

  integer                       :: EpsilonSwitch,KroneckerDelta
  double precision              :: kappa

  kappa = dble(EpsilonSwitch(i,j)*KroneckerDelta(i,k) - EpsilonSwitch(k,i)*KroneckerDelta(i,j))

end function KappaCross

!------------------------------------------------------------------------

subroutine CalcInv3(a,det)

! Calculate the inverse and the determinant of a 3x3 matrix

  implicit none

  double precision,intent(inout)  :: a(3,3)
  double precision, intent(inout) :: det
  double precision                :: b(3,3)
  integer                         :: i,j

  det = a(1,1)*(a(2,2)*a(3,3)-a(2,3)*a(3,2)) &
      - a(1,2)*(a(2,1)*a(3,3)-a(2,3)*a(3,1)) &
      + a(1,3)*(a(2,1)*a(3,2)-a(2,2)*a(3,1))

  do i=1,3
    b(i,1) = a(i,1)
    b(i,2) = a(i,2)
    b(i,3) = a(i,3)
  enddo

  a(1,1) = b(2,2)*b(3,3) - b(2,3)*b(3,2)
  a(2,1) = b(2,3)*b(3,1) - b(2,1)*b(3,3)
  a(3,1) = b(2,1)*b(3,2) - b(2,2)*b(3,1)

  a(1,2) = b(1,3)*b(3,2) - b(1,2)*b(3,3)
  a(2,2) = b(1,1)*b(3,3) - b(1,3)*b(3,1)
  a(3,2) = b(1,2)*b(3,1) - b(1,1)*b(3,2)

  a(1,3) = b(1,2)*b(2,3) - b(1,3)*b(2,2)
  a(2,3) = b(1,3)*b(2,1) - b(1,1)*b(2,3)
  a(3,3) = b(1,1)*b(2,2) - b(1,2)*b(2,1)

  do i=1,3
    do j=1,3
      a(i,j) = a(i,j)/det
    enddo
  enddo

end subroutine CalcInv3

!------------------------------------------------------------------------

subroutine CalcInv4(a,det)

  implicit none

  double precision,intent(inout) :: a(4,4)
  double precision,intent(inout) :: det
  double precision               :: b(4,4)
  integer                        :: i,j

  det = a(1,1)*(a(2,2)*(a(3,3)*a(4,4)-a(3,4)*a(4,3))  &
               -a(2,3)*(a(3,2)*a(4,4)-a(3,4)*a(4,2))  &
               +a(2,4)*(a(3,2)*a(4,3)-a(3,3)*a(4,2))) &
      - a(1,2)*(a(2,1)*(a(3,3)*a(4,4)-a(3,4)*a(4,3))  &
               -a(2,3)*(a(3,1)*a(4,4)-a(3,4)*a(4,1))  &
               +a(2,4)*(a(3,1)*a(4,3)-a(3,3)*a(4,1))) &
      + a(1,3)*(a(2,1)*(a(3,2)*a(4,4)-a(3,4)*a(4,2))  &
               -a(2,2)*(a(3,1)*a(4,4)-a(3,4)*a(4,1))  &
               +a(2,4)*(a(3,1)*a(4,2)-a(3,2)*a(4,1))) &
      - a(1,4)*(a(2,1)*(a(3,2)*a(4,3)-a(3,3)*a(4,2))  &
               -a(2,2)*(a(3,1)*a(4,3)-a(3,3)*a(4,1))  &
               +a(2,3)*(a(3,1)*a(4,2)-a(3,2)*a(4,1)))
  do i=1,4
    b(1,i) = a(1,i)
    b(2,i) = a(2,i)
    b(3,i) = a(3,i)
    b(4,i) = a(4,i)
  enddo

  a(1,1) =  b(2,2)*(b(3,3)*b(4,4)-b(3,4)*b(4,3))-b(2,3)*(b(3,2)*b(4,4)-b(3,4)*b(4,2))+b(2,4)*(b(3,2)*b(4,3)-b(3,3)*b(4,2))
  a(2,1) = -b(2,1)*(b(3,3)*b(4,4)-b(3,4)*b(4,3))+b(2,3)*(b(3,1)*b(4,4)-b(3,4)*b(4,1))-b(2,4)*(b(3,1)*b(4,3)-b(3,3)*b(4,1))
  a(3,1) =  b(2,1)*(b(3,2)*b(4,4)-b(3,4)*b(4,2))-b(2,2)*(b(3,1)*b(4,4)-b(3,4)*b(4,1))+b(2,4)*(b(3,1)*b(4,2)-b(3,2)*b(4,1))
  a(4,1) = -b(2,1)*(b(3,2)*b(4,3)-b(3,3)*b(4,2))+b(2,2)*(b(3,1)*b(4,3)-b(3,3)*b(4,1))-b(2,3)*(b(3,1)*b(4,2)-b(3,2)*b(4,1))

  a(1,2) = -b(1,2)*(b(3,3)*b(4,4)-b(3,4)*b(4,3))+b(1,3)*(b(3,2)*b(4,4)-b(3,4)*b(4,2))-b(1,4)*(b(3,2)*b(4,3)-b(3,3)*b(4,2))
  a(2,2) =  b(1,1)*(b(3,3)*b(4,4)-b(3,4)*b(4,3))-b(1,3)*(b(3,1)*b(4,4)-b(3,4)*b(4,1))+b(1,4)*(b(3,1)*b(4,3)-b(3,3)*b(4,1))
  a(3,2) = -b(1,1)*(b(3,2)*b(4,4)-b(3,4)*b(4,2))+b(1,2)*(b(3,1)*b(4,4)-b(3,4)*b(4,1))-b(1,4)*(b(3,1)*b(4,2)-b(3,2)*b(4,1))
  a(4,2) =  b(1,1)*(b(3,2)*b(4,3)-b(3,3)*b(4,2))-b(1,2)*(b(3,1)*b(4,3)-b(3,3)*b(4,1))+b(1,3)*(b(3,1)*b(4,2)-b(3,2)*b(4,1))

  a(1,3) =  b(1,2)*(b(2,3)*b(4,4)-b(2,4)*b(4,3))-b(1,3)*(b(2,2)*b(4,4)-b(2,4)*b(4,2))+b(1,4)*(b(2,2)*b(4,3)-b(2,3)*b(4,2))
  a(2,3) = -b(1,1)*(b(2,3)*b(4,4)-b(2,4)*b(4,3))+b(1,3)*(b(2,1)*b(4,4)-b(2,4)*b(4,1))-b(1,4)*(b(2,1)*b(4,3)-b(2,3)*b(4,1))
  a(3,3) =  b(1,1)*(b(2,2)*b(4,4)-b(2,4)*b(4,2))-b(1,2)*(b(2,1)*b(4,4)-b(2,4)*b(4,1))+b(1,4)*(b(2,1)*b(4,2)-b(2,2)*b(4,1))
  a(4,3) = -b(1,1)*(b(2,2)*b(4,3)-b(2,3)*b(4,2))+b(1,2)*(b(2,1)*b(4,3)-b(2,3)*b(4,1))-b(1,3)*(b(2,1)*b(4,2)-b(2,2)*b(4,1))

  a(1,4) = -b(1,2)*(b(2,3)*b(3,4)-b(2,4)*b(3,3))+b(1,3)*(b(2,2)*b(3,4)-b(2,4)*b(3,2))-b(1,4)*(b(2,2)*b(3,3)-b(2,3)*b(3,2))
  a(2,4) =  b(1,1)*(b(2,3)*b(3,4)-b(2,4)*b(3,3))-b(1,3)*(b(2,1)*b(3,4)-b(2,4)*b(3,1))+b(1,4)*(b(2,1)*b(3,3)-b(2,3)*b(3,1))
  a(3,4) = -b(1,1)*(b(2,2)*b(3,4)-b(2,4)*b(3,2))+b(1,2)*(b(2,1)*b(3,4)-b(2,4)*b(3,1))-b(1,4)*(b(2,1)*b(3,2)-b(2,2)*b(3,1))
  a(4,4) =  b(1,1)*(b(2,2)*b(3,3)-b(2,3)*b(3,2))-b(1,2)*(b(2,1)*b(3,3)-b(2,3)*b(3,1))+b(1,3)*(b(2,1)*b(3,2)-b(2,2)*b(3,1))

  do i=1,4
    do j=1,4
      a(i,j) = a(i,j)/det
    enddo
  enddo

end subroutine CalcInv4