!****************************************************
!*   Program to demonstrate Hermite coefficients    *
!* ------------------------------------------------ *
!* Reference: BASIC Scientific Subroutines, Vol. II *
!* By F.R. Ruckdeschel, BYTE/McGRAWW-HILL, 1981 [1].*
!*                                                  *
!*            F90 Version by J.-P. Moreau, Paris.   *
!*                    (www.jpmoreau.fr)             *
!* ------------------------------------------------ *
!* SAMPLE RUN:                                      *
!*                                                  *
!* Hermite polynomial coefficients for order 2      *
!*                                                  *
!*   A( 0) =        -2                              *
!*   A( 1) =         0                              *
!*   A( 2) =         4                              *
!*                                                  *
!* Hermite polynomial coefficients for order 3      *
!*                                                  *
!*   A( 0) =         0                              *
!*   A( 1) =       -12                              *
!*   A( 2) =         0                              *
!*   A( 3) =         8                              *
!*                                                  *
!* Hermite polynomial coefficients for order 4      *
!*                                                  *
!*   A( 0) =        12                              *
!*   A( 1) =         0                              *
!*   A( 2) =       -48                              *
!*   A( 3) =         0                              *
!*   A( 4) =        16                              *
!*                                                  *
!* Hermite polynomial coefficients for order 5      *
!*                                                  *
!*   A( 0) =         0                              *
!*   A( 1) =       120                              *
!*   A( 2) =         0                              *
!*   A( 3) =      -160                              *
!*   A( 4) =         0                              *
!*   A( 5) =        32                              *
!*                                                  *
!* Hermite polynomial coefficients for order 6      *
!*                                                  *
!*   A( 0) =      -120                              *
!*   A( 1) =         0                              *
!*   A( 2) =       720                              *
!*   A( 3) =         0                              *
!*   A( 4) =      -480                              *
!*   A( 5) =         0                              *
!*   A( 6) =        64                              *
!*                                                  *
!* Hermite polynomial coefficients for order 7      *
!*                                                  *
!*   A( 0) =         0                              *
!*   A( 1) =     -1680                              *
!*   A( 2) =         0                              *
!*   A( 3) =      3360                              *
!*   A( 4) =         0                              *
!*   A( 5) =     -1344                              *
!*   A( 6) =         0                              *
!*   A( 7) =       128                              *
!*                                                  *
!* Hermite polynomial coefficients for order 8      *
!*                                                  *
!*   A( 0) =      1680                              *
!*   A( 1) =         0                              *
!*   A( 2) =    -13440                              *
!*   A( 3) =         0                              *
!*   A( 4) =     13440                              *
!*   A( 5) =         0                              *
!*   A( 6) =     -3584                              *
!*   A( 7) =         0                              *
!*   A( 8) =       256                              *
!*                                                  *
!* Hermite polynomial coefficients for order 9      *
!*                                                  *
!*   A( 0) =         0                              *
!*   A( 1) =     30240                              *
!*   A( 2) =         0                              *
!*   A( 3) =    -80640                              *
!*   A( 4) =         0                              *
!*   A( 5) =     48384                              *
!*   A( 6) =         0                              *
!*   A( 7) =     -9216                              *
!*   A( 8) =         0                              *
!*   A( 9) =       512                              *
!*                                                  *
!* Hermite polynomial coefficients for order 10     *
!*                                                  *
!*   A( 0) =    -30240                              *
!*   A( 1) =         0                              *
!*   A( 2) =    302400                              *
!*   A( 3) =         0                              *
!*   A( 4) =   -403200                              *
!*   A( 5) =         0                              *
!*   A( 6) =    161280                              *
!*   A( 7) =         0                              *
!*   A( 8) =    -23040                              *
!*   A( 9) =         0                              *
!*   A(10) =      1024                              *
!*                                                  *
!****************************************************
! Explanations
! ------------
!
! Hermite polynomials are defined over the range -inf. < x < inf. The weight- 
! ing function is w(x) = e^(-x^2): 
!
!    inf.
!    Sum   e^(-x^2) H (x) H (x) dx = 0     for n <> m         (3.9.8)
!    -inf.           n     m
!                                  = f(n)  for n = m
!
! The corresponding recursion relation is
! 
!    H   (x) = 2x H (x) - 2n H   (x)                          (3.9.9)
!     n+1          n          n-1
! 
! where H (x) = 1  and  H (x) = 2x 
!        0               1
! 
! As with the other polynomials, a simple subroutine for evaluating the coefficients 
! be written (see program HERMITE). 
! 
! Note that Hermite polynomials are either even or odd, depending on N, and the co- 
! efficients are integers. 
!-----------------------------------------------------------------------------------
PROGRAM Hermite

real*8  A(0:10)
real*8  B(0:10,0:10)
integer n,k

   print *,' '
   do n = 2, 10
     write(*,25) n

     call Hermite_Coeff(n,A,B)

     do k = 0, n  
       write(*,50) k, A(k)  
     end do 
     print *,' '
     if (n<10) pause ' <Enter> to continue...'
   end do
   print *,' '

   stop
25 format(' Hermite polynomial coefficients for order',i2/)
50 format('   A(',i2,') = ',f10.0)

end

!**************************************************
!* Hermite polynomial coefficients evaluation by  *
!* means of recursion relation. The order of the  *
!* polynomial is n. The coefficients are returned *
!* in A(i).                                       *
!**************************************************
Subroutine Hermite_Coeff(n,A,B)  
  integer i,j,n
  real*8  A(0:10), B(0:10,0:10)
  !Establish l0 and l1 coefficients
  B(0,0)=1.d0 ; B(1,0)=0.d0 ; B(1,1)=2.d0
  !Return if order is less than two
  if (n>1) then
    do i = 2, n
      B(i,0)=-2.d0*(i-1)*B(i-2,0)
      do j = 1, i
        !Basic recursion relation
        B(i,j)=2.d0*B(i-1,j-1)-2.d0*(i-1)*B(i-2,j)
      end do
    end do
    do i = 0, n
      A(i)=B(n,i)
    end do
  end if
  return
end

! End of file hermite.f90