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! Copyright (C) 2002-2005 J. K. Dewhurst, S. Sharma and C. Ambrosch-Draxl.
! This file is distributed under the terms of the GNU Lesser General Public
! License. See the file COPYING for license details.
!BOP
! !ROUTINE: stheta_mp
! !INTERFACE:
real(8) function stheta_mp(n,x)
! !INPUT/OUTPUT PARAMETERS:
! n : order (in,integer)
! x : real argument (in,real)
! !DESCRIPTION:
! Returns the smooth approximation to the Heaviside step function of order
! $N$ given by Methfessel and Paxton, {\it Phys. Rev. B} {\bf 40}, 3616
! (1989),
! $$ \tilde\Theta(x)=1-S_N(x) $$
! where
! \begin{align*}
! S_N(x)&=S_0(x)+\sum_{i=1}^N \frac{(-1)^i}{i!4^n\sqrt\pi} H_{2i-1}(x)
! e^{-x^2},\\
! S_0(x)&=\frac{1}{2}(1-{\rm erf}(x))
! \end{align*}
! and $H_j$ is the $j$th-order Hermite polynomial. This procedure is
! numerically stable and accurate to near machine precision for $N\le 10$.
!
! !REVISION HISTORY:
! Created April 2003 (JKD)
!EOP
!BOC
implicit none
! arguments
integer, intent(in) :: n
real(8), intent(in) :: x
! local variables
integer i
real(8), parameter :: sqpi=1.7724538509055160273d0
real(8) sum,t1
! external functions
real(8) factnm,hermite,erf
external factnm,hermite,erf
if (n.lt.0) then
write(*,*)
write(*,'("Error(stheta_mp): n < 0 : ",I8)') n
write(*,*)
stop
end if
if (n.gt.10) then
write(*,*)
write(*,'("Error(stheta_mp): n out of range : ",I8)') n
write(*,*)
stop
end if
if (x.lt.-12.d0) then
stheta_mp=0.d0
return
end if
if (x.gt.12.d0) then
stheta_mp=1.d0
return
end if
sum=0.5d0*(1.d0-erf(x))
do i=1,n
t1=1.d0/(factnm(i,1)*dble(4**i)*sqpi)
if (mod(i,2).ne.0) t1=-t1
sum=sum+t1*hermite(2*i-1,x)*exp(-x**2)
end do
stheta_mp=1.d0-sum
return
end function
!EOC
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